Try doing a "double-fork" to detach the long-running process from the Web server. Something like:

if (my $pid1 = fork())
{
  wait;
  return;
}
elsif (defined($pid1))
{
  # Child
  if (my $pid2 = fork())
  {
    # Still Child
    exit(0);
  }
  elsif (defined($pid2))
  {
    # Grandchild
    exec(...);
  }
  else
  {
    die "Fork error: $!\n";
  }
}
else
{
  die "Fork error: $!\n";
}
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You may also want to do things like POSIX::setsid and setpgrp. Search the archives for daemon for some sample code.