I am currently ploughing through Piers Anthony's "Split Infinity", which involves the adventure of a "Game player" transported from his home universe planet of "Proton" (MacGuffins galore) which seems to be in a 1-to-1 correspondence with an alternative universe - so far un-named. I
suspect that this is the universe in which Anthony's "Adept" books, as well as "Phaze" are set - none of which I've read, but I've seen what look very much like cross-references in "Split Infinity" in the first 100-odd pages.
Taking "Phaze" as the name of this "Split Infinity" universe, I propose this as another "dimension". It isn't terribly good on the "Everybody Knows" front, but at least some people in the protagonist's original universe know about the alternate universe, and how to get there, so it at least meets the "colsely held secret" level of knowledge.
I suppose I'd better see what spoilers Wikipedia has ... Ah, the hive-mind there does
cluster "Split Infinity" with several other "Phaze" and "Adept" books, but no name for the alternative universe.
Never having been a fan of PA's "Xanth" 40-odd, punishing books, I'm not sure if they qualify. I have a sneaking suspicion that they do, vaguely remembering some interaction between Xanth and our nominal Earth ; Xanth itself has some sort of relationship with PA's home state of Florida.
His "Incarnations of Immortality" series has a nominal Earth dimension, a "heaven", a "hell", and a "purgatory", between which dimensions the titular Incarnations yo-yo with gay abandon.
Is Piers Anthony culled, like an inhabitant of "The Barn"?