Civil war, logistics, and you(r location)
Cities are left wing. Rural is right wing. It would be a war on infrastructure, not a war on people. The right will, if it is smart, do its best to disrupt the logistics required to maintain the huge population density of cities. The left will play defense. If cities go into Venezuela mode because supply lines are cut, the leftist force will be split between peacekeeping on its own turf and trying to defend whatever critical infrastructure is left. Diversity in cities will cause factions to form along racial lines (or ideological lines if among ideological fanatics) as is the case in prisons and everywhere else you have a population isolated within adverse conditions. Tribalism takes hold. Every country is four meals away from revolution, and this logic would apply to each isolated city. Honestly, this whole situation is a nightmare for the left/urbanites. That would be the overall conflict.
There will of course be some armed conflict. But modern weaponry and tactics almost require asymmetric warfare at this point in time. If a militia/warlord becomes too large or powerful, it becomes a target for expensive, high tech ordinance, which I am sure both sides will have. So you will probably see a situation like Iraq after "Mission Accomplished" - right and left wing militias, both of which have military weapons, support and training (like the Ba'athist Suuni vs the Shia supported by international elements). The main difference will be the schism within the US military/intelligence community, rather than between the prior government and the 'liberators'.
Logistical issues will be faced by both sides. For the right, it would be absolutely critical to obtain Houston for fuel. The left will have plenty of fuel due to their costal location and multiple ports. For the left, food and water are the mission critical items. Most American produce is grown in Mexico and California, and travels through half the country by way of the Greenville Bridge and Big River Crossing in Missouri. If this route is broken then the eastern seaboard loses a sizable percentage of its food supply. Then you have cities like NYC and others where a huge population is connected to the rest of the country by a few main arteries. If you sever one or two the rest of them will become impossibly congested for two reasons. One, increased traffic. Two, increased security to make sure none of the remaining entryways are destroyed.
Just to put this into perspective NYC has a population of 8.8 million. Let's say for a number, people need to eat 2 pounds of food a day. The weight limit for a semi truck payload is 45k pounds. That means each truck can feed 22.5l people per day (probably closer to 20k because of packaging weight, etc). That means you need to get about 400 trucks inspected and across the network of bridges/tunnels. This would be easy, but not when you also have civilian traffic. So the government/whoever is in power then must create designated shipping routes which are off limits to civilian traffic. This makes logistics easier, but it also makes you a giant glowing target for both right wing insurgents and the hungry. You also have long lines of trucks concentrated alone one route. If there are 27 feet between trucks you have each truck taking up 100 feet. That is a 7.5 mile long line of trucks trying to get access to NYC which require inspections to make sure there isn't a bomb hidden in your trailer full of potatoes that would blow yet another bridge and further complicate this logistical nightmare.
The next logistical issue is, the trucks have to make it from the center of the country, all the way to the coastline to keep the left wing fed. Surprise surprise, its not in the developed left wing centers that have farms. We are also a net exporter of food (1) so the rest of the world might not have quite the supply necessary to sustain our most densely populated areas. Another point to consider is the countries we import most from are Mexico and Canada, both of which are separated from many major metropolitan areas by vast swaths of red territory. When you couple this with the fact that trucks have 200 gallons of diesel fuel in the tank (going back to the need for petroleum earlier) and that they probably have food or other necessary supplies - they become a very attractive target for economic reasons as well.
Going into how the military schism might happen is not only too chaotic to predict but also several paygrades above me. However, Millennium Challenge 2002 is a wargames exercise that unpacks some of this. That being said, basically the right wing force has a huge upper hand in this. As long as local groups get together and fuck up roads and bridges (ancap paradise) cities simply won't be able to provide for their huge populations well enough to avoid their own rebellions and violence. You can't hunt deer in Central Park. You can in the Midwest.
This is if the right wing militias only target roads, bridges and semitrucks. If communications, the power grid and water-treatment plants are targeted, chaos would reign supreme in any city over 250k people. Meanwhile, Bubba and Jim can live off corn, potatoes, beans, deer and groundhog pretty easily. They also have fresh water that isn't polluted by the industry and garbage of a city, and wood to build a fire for boiling. So basically, it comes down to grab SKS, go innawoods.
About that military schism - if one sub commander decides to go full right wing victory mode, all he has to do is torpedo a few of those mega container ships and the whole shipping industry will lose its mind. You are looking at hundreds of millions of dollars in damage. How much food can you divert from the rest of the world to America? This is a zero sum game, there will be a large surplus of food left on farms to rot as it is not allowed to meet its customers. Europe isn't that large and they have a high population. Africa can't even support itself agriculturally, same with China and India. The Middle East isn't exactly in a position to grow anything as we've been bombing them for the past seventeen years off and on. This leaves us with South America as our primary source of agricultural products, South America (Brazil and Argentina primarily) mostly exports to Europe (2) with the European Union being its biggest customer while China was the largest single importing country with a 13.2% share, followed by the Netherlands with 9.5% and the US at 8.7%. Again, how much of this can they divert to coastal cities with these huge populations? Rich people will of course have whatever they want, but the poor people may decide to take it by force is the point here.
I wouldn't anticipate a naval blockade, as most Americans don't want to starve their own people to death. Maybe if you have a right wing naval force they will hold the container ships as a bargaining chip. I'd anticipate a few right wing yahoos going full pirate mode, just because there are 259 coast guard cutters and 296 battle force ships in the navy. You know there will be a few.
It would become the greatest proxy war of all time. Every country would have a huge interest. I still see it as being concentrated around the coastal cities though as the center of the country either has (mostly) right wing cities which have blue pockets of people who would be useless in a warzone and the people (of color) they're terrified of. When a civil war type scenario becomes a reality in such locations, there will be groups that won't tolerate criminal gangs and will deal with them using extreme prejudice. Chicago and Detroit would probably see a total white flight and fall into chaos with a huge death toll. Once resources in such areas are depleted and trucks of bloods or crips show up into rural Missouri to look for food, well, they'd stick out like a sore thumb and probably disappear pretty quickly.
Back to the proxy war idea, China has a greater interest in American agricultural exports than anything else, and have already stolen most of our defense research already (3). So the real quandary for both the Chinese and the Russians is how do you support the right wing elements in America if they don't want to work with you? The right wing are the pro-America bunch. Many of them are boomers who are armed to the teeth and lived through the Cold War. They hate communists, hence their hatred of left wingers. What I would expect to see is more of the same. Hacking. Disabling communications, the power grid, overall disruption of the high tech necessities facilitating the life of urban centers. Without the internet, social media, television and electricity, urban life would be completely destabilized for two main reasons. First, people are now content with sitting in one room absorbing media. Without digital entertainment, they're just sitting alone in a room. The streets would be flooded with people who no longer know what to do with themselves. It would take three days for things to turn into countless versions of that Twilight Zone episode The Monsters Are Due On Maple Street.
Second, without electricity you cannot cook or boil water. You have no heat or air conditioning. You have no light at night. People in rural settings have woods with dead trees which to make fires. You can only burn office furniture for so long and I certainly wouldn't cook on it. Our grid is 100 skilled people away from total chaos. You think China wouldn't send 1,000 skilled people, in addition to using every hacking trick in the book to disrupt our power and communications? The only thing keeping them from launching a full on assault on us is our nuclear deterrent. I see them as being a disruptive force sent to sabotage. Russia would be similar, but it would be easier for them to blend in. There are plenty of Polish immigrants in the Midwest they could disguise themselves as. They would be the group to act as our green berets do, training the right wing militias. I don't see either China or Russia aiding the left wing. They know our financial industry is all about fucking them over and they would like to see it crash and burn as much as the right wing would.
Israel would be the one to look out for. There are enough neocon Christians out there that may still believe the "greatest ally" bullshit, and you know Mossad will be playing both sides in this conflict. Ultimately they will want the left to win, but only once they get them to stop allowing criticism of Israel's treatment of Palestinians. The EU will unsurprisingly fully support the left wing. This will be logistical support though since, as I've said, this will be primarily a logistical conflict. It really comes down to the Navy's role in this scenario. If shipping lanes are cut off, this would be a very short affair. If shipping lanes remain open, it would be years of conflict. The right will eventually win, the only question is what the terms of peace will be.
Another factor which must be considered is the sentiment within cities. The people who are going to have all the critical power and be doing all the work in cities are very much blue collar. Dock workers, linemen, telecom workers, truckers, policemen (especially policemen), firefighters, doctors (not blue collar but usually pretty right wing after treating the population at large) and construction workers. They are going to be working their asses off with longer hours, the threat of violence and no guarantee of food or security. As time progresses the blue collar workers morale will be constantly decreasing,. If these people decide the right wing people are right, the left's position crumbles. A financial analyst and a lawyer don't know how to operate a crane. A yoga instructor can't troubleshoot a high voltage distribution system. I also doubt that a social worker or marketing executive or anyone in HR can deal with people looting the local grocery store. This doesn't even take into account the possibility that Bill the heavy equipment mechanic might loosen a chain on the drive system for the port's cargo container crane causing it to break (with a 6 month repair time because the factory that makes the part is in Minnesota and logistics throughout the country are fucked). If you were a trucker, would you risk your life to bring stuff to people who you don't like? Cities simply don't have anything to offer but money, and I am willing to bet most things outside the city will be pretty darn cheap as there will be no money supply (banks/debt) and a surplus of food which isn't going to cities because of the disruption. The economy is going to be totally split if this happens. The total consequence of this will be difficult to determine but cities will probably see hyperinflation and rural setting will probably see deflation but mostly a return to trade and barter.