Scented with cologne; having had cologne applied to one's body or clothing.
From 'cologne' (eau de cologne, the fragrant water originally made in Cologne, Germany) + the past participle '-ed.' The term describes something treated or perfumed with this scent.
Cologne was actually named after the German city where it was manufactured, so 'cologned' literally means 'treated with that water from Cologne'—it's a funny reminder that we often don't realize product names are just place names.
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