Q. Chicago says commas aren’t needed with “not only . . . but also” constructions but are needed between two independent clauses joined by a coordinating conjunction. So which wins when you have both?
A. When a pair of correlative conjunctions like not . . . but, either . . . or, or both . . . and joins two independent clauses, the clauses should be separated by a comma unless they are very short and closely related. In other words, the rule about independent clauses takes precedence. See CMOS 6.22 and 6.49 for more details.