I've cut the handler down as much as I think I can to just:

  while ((my $pid = waitpid(-1, &WNOHANG)) > 0) {
    $children--;         # need this to tell when we're done
    $exits{$pid} = $?;   # need this to see the problem
  }
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then I dump %exits at the end of the script. However, I still see output like:

18106 0
18116 0
18113 -1
18105 0