Characters: Emishi Haruki and Clayman
Fandom(s): Get Backers
Spoilers For: Manga up to the Hide and Seek arc, anime up to the Venus de Milo arc
The Evidence: We know very little about Clayman. We don't even know her real name. However, what we do know about her is that she loves art a lot (which is actually an understatement), she values her mother's legacy quite a bit, and she is friendly and soft-spoken as well as highly intelligent and not above outsmarting people simply to prove a point. The manga adds that she went to art school in France, where she met the only friend we know she has: Kite, the genius artist who met his downfall when he became addicted to a potent form of heroin and became obsessed with the Venus de Milo, causing him to reject his own fiancée. Clayman is a solitary person, but she doesn't seem very happy about this arrangement: while she understands that being a wanted criminal limits the number of people she can let get close to her, she still invites Himiko to join her, asks Ban to visit her more often, and is still in touch with Kite (who she took care of after his drug habit forced him into a catatonic state until he was reunited with his lost love, Hera).
Clayman also seems to have a deeper understanding of the feelings of others, especially artists, than the rest of the cast: her concern over the paintings in her collection comes from her understanding of how much of their soul the artists put into their last works. She dislikes the thought of art being squandered away or destroyed, and wants the rest of the world to enjoy it on the same level that she does. Although her obsession with art seems to consume her, the end of the Venus de Milo arc proves that she understands that there are things in this world more important than art, and perhaps caused her to entertain thoughts of seeking companionship or dwell on feelings she already had.
Emishi Haruki, however, seems like a rather unlikely companion for someone like Clayman. He's loud, random, and silly. He is not very well educated, but can compensate with some street sense he gained from living in Mugenjyou from an early age. He is as much warrior as he is comedian, and is passionate when it comes to seeing to both the safety and happiness of those that he cares about. Danger is nothing new to him, and as a result someone leading a dangerous lifestyle won't make him bat an eye.
Emishi likes women, and makes no secret of this. He doesn't have much luck with them, however, probably in part to his very unusual sense of humor (his jokes are funny only in that they are pathetic, but he has a talent for making the most random actions ever). But when he honestly likes a girl, he knows how to act like a perfect gentleman (even if he makes lewd comments to himself in the meantime). He sees to respect that women can be his equal, and when they aren't he's willing to protect them.
Also, Emishi may have a large group of people he cares about, but he has only been truly close to Fuyuki Shido and Natsuki Amon, the latter of which died (and lead to a serious depression for quite a while afterwards, leading to him making advances on Sakura). He has witnessed one of his best friends find true love, and has professed to being a romantic himself.
Conclusions (a.k.a. Why They Should Meet): Clayman and Emishi both seem to be at a place in their lives where they want companionship. Likewise, Emishi's friendly, devoted nature is perfect for someone who is solitary and reserved. One gets the feeling that Clayman's life hasn't had much humor in it thus far, and she may actually be one of the few people who finds Emishi's bad jokes genuinely funny. Likewise, Clayman's intelligence and refinement is the perfect foil for Emishi's decided lack thereof: she is patient and understanding enough to not overshadow him or make him feel stupid, while helping him to understand and learn about her world. Emishi living a life used to danger would also make him an ideal companion for someone who may worry about putting other people in danger.