I have a hard time keeping up with anime couples, but as I wreck my brain to remember couples I don't like or who make a poor pairing I came up with these two pairings:
Light and Misa from "Death Note". I haven't seen this anime is years but I recently watched the Netflix live action movie and I just remembered I didn't like this pairing from the anime. Also both characters were psycho.
Eren and Mikasa from "Attack on good taste". I just started watching the latest season of the anime and I like the characters Eren & Mikasa but I just don't like them as a couple. Also I get this feeling that Mikasa is in love with Eren but he doesn't feel the same way about her. I also dislike one sided romances.
What is real? How do you define 'real'? If you're talking about what you can feel, what you can smell, what you can taste and see, then 'real' is simply electrical signals interpreted by your brain.
Joseph might not like this one - 02 and Hiro. They're, like, so in love! Like deeply! You guys. SO in love. <3 <3 <3 But for real though - it's a relationship with a really teenage feel, and it brings alllll the requisite drama, complete with dragging the friend group into it. Also it's pretty unfortunately codependent, but I guess when you make your analogy of couplings out of mythological 1-winged birds that can't fly until they find their symmetric match, that's kinda what you get. There are parts of their realtionship that are also kind of touching, yes; but SO MUCH drama.
Admittedly, I can't think of a ton of really loathsome couples, and that's probably because so many titles make alot of content out of "will they or won't they" and often even when "they do" that's the grand finale, and you never get to see their coupledom play out to the good or bad. At least Hiro and 02 managed to make it to canon status.