Always registered, always appear correctly, just that they’re being hidden by the ACF JavaScript when the page is being edited.
However, upgrading to 5.7.12 seems to have fixed this issue. I couldn’t see anything specific in the change logs, but the function that was doing this seems to have been considerably re-written compared to 5.7.7, so this bug has been addressed (either intentionally or unintentionally!).
Thanks for your efforts.