What's the anime where a girl who comes to town to fight an enemy meets a guy who lives with an old man? (It's not Bleach)

I vaguely remember parts of it, and I’m not sure about the exact details, but something along the lines of:

• the guy(MC I think ) is an orphan but lives with an old man who teases him (in a monastery?)
• the girl has long black(?) hair with bangs, has a sword, is part of an organization(?)
• she remembers in a flashback how an old woman once told her when she was a child that she’d find someone someday who’d love her to death
• anyway she came to town to fight this great evil(a demon?)
• fighting in general happens by grid and on rooftops
• they meet somehow after a fight & she kinda falls in love?
• she thinks it’s this guy who’s her soulmate & she doesn’t wanna lose him
• later something happened and she willingly went to fight this enemy alone, who ended up taking control of her
• one particular scene goes like this: the main character goes looking for her & ends up finding her with her eyes glowing purple, there was lightning in the background…she tells him to go back & there’s nothing he can do or something like that

That’s all I can remember. I saw it 8 or 9 years ago. Please, this anime has been bugging me a lot lately, and I can’t figure it out.