 If you liked
Kamisama Dolls
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 ...then you might like
RDG: Red Data Girl
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In both series, the MC moves to a new place to seemingly begin their new life but strange events happens around them in supernatural form.
There is indeed supernatural themes in both series that also involves a higher power. These powers seemingly surround the origins of both series as the main protagonist is the center of it all. There is also some mystery phenomenon in both series and hinted romance.
Among other factors, Kamisama Dolls and Red Data Girl has a similar vibe with their settings in the modern world but with some traditional backgrounds set in previous times.