In a manner relating to or affected by gout, or in a way that shows signs of gouty illness.
From 'gouty' (adjective, meaning affected by gout) plus the adverbial suffix '-ly'. Gouty itself comes from 'gout' (from Old French 'goute', meaning a drop, because the disease was thought to result from a drop of humor flowing into the joints).
Medieval physicians thought gout came from literal 'drops' of bad bodily fluid dripping into your joints—that's why the French called it 'goute' (drop). The name stuck even though we now know it's about uric acid crystals, not mystical humors!
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