I tried closing STDOUT, STDIN and STDERR with no apparent difference. A further oddity I've noticed is that when it does actually finish (by running a 'long' program that isn't really all that long), the page returned by Apache from the CGI contains two copies of the message, though only one copy of debug messages in the log.

My code now looks like this, including the bit of code before the extract I showed

        binmode(STDOUT, ":unix"); # unbuffer STDOUT

        print $q->header(
            -status   => '200 Pages Updated',
            -type     => 'text/plain',
            -charset  => 'ascii',
        );

        print "\n\nStarting Update of $app_name\n\n"
            . "Since this will take a while, we will send a mail "
            . "when it is complete\n";

        warn "Forking '@cmd'\n" if $DEBUG;

        use Proc::Daemon;

        my $daemon = Proc::Daemon->new(
                     );

        close STDIN;
        close STDOUT;
        close STDERR;

        my $child_pid = $daemon->Init;

        if ($child_pid)
        {
            # We are the parent - we need to exit so Apache shows what
            # we just printed to the user

#            warn "Child PID is $child_pid\n" if $DEBUG;
            exit;
        }

        my $status  = system @cmd;


        # send an email after completion instead of updating the page
[download]

The 'Starting Update...' text appears twice on the HTML page, but the 'Forking...' message only appears once in the log. A single email message is sent as well.