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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).
This situation arises when the subject of the first clause follows rather than precedes the opening conjunction. In that case, another subject (often the same one) is required after the coordinating conjunction, which is then preceded by a comma:
Not only did we camp in the woods, but we also canoed down the river.
Either we will camp in the woods, or we will canoe down the river.
Compare the same sentences but with the subject moved to precede the opening conjunction:
We not only camped in the woods but also canoed down the river.
We will either camp in the woods or canoe down the river.
In that case, the subject isn’t repeated, so no comma is needed (see CMOS 6.23 and 6.46). For more information about correlative conjunctions, see CMOS 5.199 and 5.244.