So... is Uragiri wa Boku no Namae wo Shitteiru finished or not?
NO, it is not finished.
Why?
The anime finished before the manga:
Anime finished: September 20, 2010
Manga finished: April 24, 2017
MangaUpdates contains information about the anime adaptation and indicates that it is incomplete.
Watching Order (by release date)
Here you have all the anime relations we detected ordered by release date, so you can see the reason for this status in context and also use it as watching order.
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Synopsis
Yuki Sakurai, an orphan with a mysterious ability to see people's painful memories, is troubled by feelings of isolation and anonymous death threats. He feels a growing unease within himself, a sense that something important is about to happen.
He is soon drawn into a world of hidden agendas and powerful protectors, marked by the arrival of a striking stranger who seems inexplicably familiar. This encounter awakens buried memories and sets Yuki on a path to uncover his true identity and the purpose of his existence.
As Yuki navigates newfound friendships and confronts potential betrayals, he must piece together the fragments of his past to understand the forces at play and the significance of his unique abilities. His journey is one of self-discovery amidst a web of intrigue and hidden truths.