SEOUL—As he entered a nightclub here recently, John Choi subjected himself to the coronavirus protocol: He wore a face mask, got his temperature checked and grabbed a pen to jot down his information at the door. But when he glanced at the sign-in sheet, he saw celebrity names that he assumed to be fake.

His fellow clubgoers had reason to be coy: They were largely from South Korea’s LGBT community, where many still conceal that side of their identities from family and colleagues because they fear they will be stigmatized in…

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