In a manner that expresses or conveys a compliment; by way of compliment.
From complimental plus the adverbial suffix -ly (from Old English -lice). This creates an adverb describing how an action relates to compliments.
This word exists in the logic of English word formation but almost never appears in actual writing or speech—it's what linguists call a 'possible but non-existent word,' perfectly grammatical yet virtually unused, showing the gap between rules and reality.
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