In a manner relating to or characterized by clairaudience; supposedly by means of hearing messages beyond normal auditory perception.
From clairaudient + -ly adverbial suffix. This layered construction (clair + audience + ent + ly) shows how English stacks morphemes to create highly specialized descriptive words.
Adding '-ly' to create adverbs is so automatic in English that we can build adverbs from almost any adjective—even rare ones like this—which is why English speakers understand 'clairaudiently' even if they've never encountered it before.
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