So... is Tayutama: Kiss on My Deity finished or not?
NO, it is not finished.
Why?
The anime finished after the manga:
Anime finished: June 22, 2009
Manga finished: May 26, 2009
... But 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
Yuuri Mito, a seemingly ordinary teenager with an unusual family tradition of exorcism, inadvertently unleashes ancient and dangerous entities known as "tayutai" after discovering and mishandling a relic.
This act releases chaos, drawing Yuuri into a conflict between these powerful beings.
Adding to the complexity, a beautiful goddess appears, declaring her intention to marry him, further complicating his already turbulent situation.
Yuuri must now navigate this unforeseen war between the tayutai while dealing with the goddess's affections, despite his own inexperience and the weaknesses of his physical form.