Yui Kanonsho: Lost Memories

Stranded in the space between human and demon, Miyu was born a vampire and charged with the responsibility of returning evil demons called Shinma to the dark. Being eternally 15, she yearns to return to the dark herself but not until she has banished the Shinma from Earth. And since her awakening, she remains cut off by the facts of who and what she is. (Source: Anime News Network)

A mysterious girl arrives at the door of a church covered in blood. She utters out the name "Nagi" before collapsing. She's taken in by the priest of the church. She starts out with no memories of her past, but slowly has strange dreams about her mother, and from that she discovers her name, Yui.