The number, gender, and person of a pronoun must match its closest antecedent. Most style manuals advise using a singular possessive pronoun when the antecedent is a disjunctive set of singular nouns:
Either FDR, Churchill, or Stalin forgot to tie his shoes before the conference.
This makes sense because the pronoun refers to any one of the three nouns.
Meanwhile, a plural pronoun is appropriate when the antecedent is a group of singular nouns joined by a conjunction:
Churchill, Stalin, and FDR attended the conference but forgot to record minutes of their deliberations.
Again, this makes sense because the pronoun must refer to all three nouns.
Question: Is a singular possessive pronoun still appropriate when the antecedent is several nouns joined with "or" but the pronoun must refer to all the nouns at once? Here is an example:
Churchill, Stalin, or FDR or one of their male staff members dressed as a woman. The KGB found a dress in the shared laundry hopper.
Using "his" as the possessive pronoun just doesn't sound right in that sentence. Is that because the pronoun is really referring to the group of Churchill, Stalin and FDR? Parenthesis help illustrate:
(Churchill, Stalin, or FDR) or (one of their staff members) dressed as a woman. The KGB found a dress in the shared laundry hopper.