I think ideally politics is a way of executing policies, which is a way of achieving goals, which are important because of values. And I think that underlying values are what is important in a relationship. There are at least three broadly coherent ways I can see people establishing a long term relationship across the political divide.
1). They aren't actually very political. There are a lot of people who don't vote, and who don't care, and/or who don't talk about politics or political stuff. My wife and I disagree vehemently on whether you should put raisins in cinnamon buns, but it didn't come up until we'd been dating for years and as long as we're on a low-carb diet and not running a bakery, it's not going to affect our relationship. But in the weeks leading up to the recent federal election, we talked about who were planning on voting for on a pretty regular basis. The Sanders voter who didn't find out his girlfriend voted for Trump until after a few months must not have talked politics very often.
2). They view politics as a team sport. This is the classic media view; it's no surprise that the classic R/D relationship was Carville and Matalin; they were both strategists, the unique breed of political animal who is only interested in policy as much as it affected the ballot box. They may as well have been ad execs for Nike and Adidas or Coke and Pepsi. And it's maybe unsurprising that the people who view politics as a team sport are either those who aren't affected by it (because of their privilege) or who just don't notice the effect (perhaps the "take your government hands off my Medicare" low-information folks.)
3). They have very specific, single-issue views. A progressive who is a PETA member hyperfocused on animal rights could plausibly find happiness with a conservative who views politics entirely through an anti-abortion lens, for instance, as long as they never move on to discussing other topics.
I think there used to be room for general moderates who shared the same basic values but differed somewhat on priorities or goals and were juuuust on the other side of any given policy from each other. Carbon taxes reduce greenhouse gas emissions but are also economically regressive, and two honest people could be opposed to both climate change and inequality but disagree on which is more important. (Pro tip: If you combine a carbon tax with a small means-based rebate or a large flat rebate, it's no longer regressive.) I think the media largely still thinks this exists.
The thing that has increasingly happened, though, is that conservatives - especially the Republicans -- have adopted as their policy "the opposite of what the Democrats want, updated daily". As Brad DeLong
put it, Obama came into office with Mitt Romney's health care policy, John McCain's climate policy and George H.W. Bush's foreign policy, and was lockstep opposed by the Republicans. In an environment like that, where policy isn't about accomplishing goals or even expressing values, I don't know how you can have a relationship across the divide, where one side of the divide is a chasm of nihilism whose only coherent, lasting value is opposing the other side.
The "rolling coal" folks are maybe the ideal example. Like, everybody used to believe that pollution was bad; some people thought that a certain level of pollution was acceptable because of the economic benefits, or because they didn't want to change the way they lived, or because they didn't like government interference. I don't think those are good arguments necessarily, but at least they're coherent. The "rolling coal" folks are actually expressing their support of pollution; they're going out of their way to cause it, entirely because it pisses liberals off.
I can understand two people having a relationship where they share value A and value B but one thinks A is more important and the other thinks B is more important. I can even understand two people sharing A and thinking it's important, and disagreeing on B but not thinking it's as important. But how do you have a long term relationship with someone whose guiding political value is "fuck you"?