iMessages asks for old Apple ID email (never had more than one account)

This problem started after the iOS update to 15.5.

I have two iPhones, a US one and a Belgian one, an iPad, and two MacBookPros that use my Apple ID. For some reason, after the iOS update, Messages on my US phone (and only there, not on any other device) started asking me to sign in with my Apple ID. Instead of the Apple ID that I'm already signed in with, it mysteriously asks for an old email that I set up my Apple ID with long ago. Since I changed the email address on my Apple ID account — I didn't create a new Apple ID with my new email — there is no Apple ID associated with that old email any more. (I've tried looking for that account to see if it exists, but the ID lookup tells me that there is no account associated with that email. That's the way it should be.) What this means is that there is no way for me to sign in with that old email to do anything about the problem. I am correctly signed into my Apple ID for Music and everything else but signed out for Messages and FaceTime.

I've tried toggling Messages and FaceTime on and off, and then selecting "Use your Apple ID for iMessage". When it asks : Use the Apple ID "current-email-ID" for iMessage, I select "Sign In", and then it takes forever to come back with: "Apple ID Verification Enter the password for “old-email-ID" in Settings." Somewhere along the way, it reverts to my now-defunct old email. I've tried selecting "Not Now" (which does nothing) and "Settings" to see if I can try my old email+password combination, but nothing works.

I've never had more than one Apple ID. I updated the email more than 7 or 8 years ago but Messages is suddenly asking me for that old email. I'm already signed in with my current email for everything else (Music, FindMe, iCloud backup — everything else works fine) but the phone mysteriously asks for the old email only on Messages. As a result, Messages on my various devices are not syncing.

Any ideas on what I can do to properly sign into Messages again with the right Apple ID email?