 If you liked
Natsume Yuujinchou
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 ...then you might like
Hyakki Yakoushou
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Both involve people who have a "gift" that enables them to view spirits (not to mention not entirely friendly spiritual "guardians"). Natsume Yuujinchou seems to be a more shoujo variation on the themes of Selected Pandemonium, which is aimed squarely at an adult audience. Both are well worth a look if you like stories about the Japanese view of the supernatural.
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Recommended by kasumisama
Both are about a teenage boy involved with Yokai. Both have a youkai/spirit as a bodyguard, and both have a grandparent that had way more spiritual power than them.
Main character is a boy who can see spirits, an ability he inherited from his grandparent. Keeps his ability secret from others. He has an antagonistic relationship with his spirit-guardian. Helps and fights spirits depending on the situation.
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Recommended by IceAndCream