I learned something new from this: I was always under the impression that you could not do anything when your process is stopped via ^Z. But apparently there are two separate signals involved here: SIGTSTP (sent when ^Z is pressed and blockable) and SIGSTOP (also suspending the process but NOT blockable) (This reminds of the time SIGCLD / SIGCHLD were different).

I have a system where shell users run commands that can grab some locks, and they like to press ^Z to suspend them which isn't compatible with a simple fcntl-based lock.