I've had success with something as simple as:

The trailing ampersand causes the process to go into the background and the system() function returns immediately. Later requests from the user can examine the contents of the logfile.

I believe mod_perl does some magic to ensure that the background process does not inherit a copy of the socket that the request came in on (if it did, the initial request would not complete until the background process exited.

If you're trying to do this on Windows, you might want to look at Win32::Process - I've used it for a similar effect.