In a manner that expresses compliments; in a praising or flattering way.
From 'complimenting' (present participle of 'compliment') + '-ly' (adverb-forming suffix). This chains together modern English's most productive suffix: verb → participle → adverb.
The '-ly' suffix is so productive that we barely notice we're doing something grammatically radical when we create adverbs—'complimentingly' is instantly understandable even though it might appear only a handful of times in English literature.
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