Say that an extension of ZFC is $\omega$-complete if any two of its $\omega$-models are elementarily equivalent. While "ZFC+V=L is $\omega$-complete" is easily disprovable in theories only slightly stronger than ZFC, I don't immediately see how to do it in ZFC alone. In particular, my question is:
Is the theory ZFC + V=L + $\omega$-Con(ZFC+V=L) + "ZFC+V=L is $\omega$-complete" consistent?
("$\omega$-Con(ZFC+V=L)" - the statement "ZFC+V=L has an $\omega$-model" - is included to avoid the trivial answer; of course, I could have written "$\omega$-Con(ZFC)" instead, but I thought this was a bit clearer.)
Note that one difficult here is the current paucity of sentences independent of ZFC+V=L which do not involve a jump in consistency strength and are not arithmetic.
This question was negatively answered at MO by user Farmer S, using $\omega_1^{CK}$-recursion theory. (I'm posting this here to move this off the unanswered queue, and have made this CW to avoid reputation gain for someone else's work.)