(I don't have any specific examples of relationship as a character, but from what I recall, How To Train Your Dragon is a film that puts a good deal of emphasis on the Relationship throughline and does it well)
As for the relationship throughline as a character... well, that's hard to picture. To me that seems like picturing the Overall Story throughline as a character. I can't think of a way to do that.
But I think the first thing you really need to figure out is: "what is a relationship???"
From my understanding (and I'm certainly not an expert), it's that hard to define space between people. The two shot example is probably the closest you can get to "showing" the relationship, but even that might be misleading, because a relationship is not about the people within the shot so much as what's going on between them.
Relationships are a lot easier to for me to visualize when I think about the change or growth of a relationship. For example, maybe two characters are forced to dance together in a weekly dance class. This leads to them growing from estranged dance partners to close friends. That's a relationship that changes from strangers to friends. I can see that arc in my head.
I also want to note again that it's important to not focus too much on the characters that are "in" the relationship. This is because relationships are not necessarily dependant on the characters and their opinions of one another. For example, I might love my dance partner and think highly of him. And he might feel the same about me. But if we stop dancing together, our relationship will never be the same. If we don't find time to reconnect and maybe try and dance together every once in a while, we'll effectively return to being strangers again.
So once you understand it's not so much about the characters, and it's more about dynamic between them, it then becomes easier to see how to work the RS throughline into your story.
It's just like how the MC throughline conflict imbalances the MC personally, and the OS throughline conflict imbalances everyone. The RS throughline conflict should imbalance the relationship. It's not about who is arguing with who, but how that arguing and conflict threatens to shatter (or save) a relationship.
In my example, I think that it would be a conflict of Doing, of not taking time to dance together, that threatens to tear the relationship apart. And it's only through Doing that the relationship can be saved.
So, I still don't quite understand relationship as a character. It certainly isn't a character in the way I typically think of characters. But I think the idea of treating it like it's own character is to separate it from the characters themselves. Stop looking at character X arguing with character Y. Start focusing on the "character" between them, the character of lovers turning into enemies.