A comma after the first John helps readers avoid the slip of reading the first part of the sentences as 'Brian moves across John and John'. We have two independent sentences joined by a coordinating conjunction. (later) Sorry, I meant to write 'two independent clauses'.
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SnarfI can't tell if "John moves back is a sentence, as you must know, and a sentence is always an independent clause.andJohn moves back" is an independent clause or not.