In english grammar, if a pronoun refers to a phrase containing every or any that falls within a different independent clause, the pronoun cannot be singular.

My question is that wheather the following sentence is right or not.

Every man left; they took their coat with them.

I should think, just from an instinctive "it doesn't sound right" kind of place, that the coat would necessarily need to be plural unless several men were able to fit into one coat?

Just a guess on my part and not to be taken as grammatical law.