In a complimentary manner; in a way that expresses praise, flattery, or free of charge.
From complimentary (from compliment plus -ary, a variant of -al) plus the adverbial suffix -ly. This creates an adverb describing how something is done in a complimentary way.
This adverb carries an interesting double meaning in English—something offered 'complimentarily' could be free (as in 'complimentary tickets') or flattering in tone. The ambiguity actually exists in the adjective 'complimentary' itself, a rare case where a word intentionally carries two meanings.
Complete word intelligence in one call. Free tier — 50 lookups/day.