This does not fit well with the basic mechanics of the Web, in which
each transaction comprises a single request and response.
If your processing can be done on the Client machine, you can use
a clientside application; for example a Java applet.

For processing on the server, one trick that works well for Clients
running an X server (and far more efficient than a JAVA solution) is:
  if ( fork() ) {
    print HTML page explaining what's going on and advising about xhost
  } else {
    exec ("xterm -display THEIR_DISPLAY -title MY_APP -e MY_PROG ARGS
        < /dev/null > /dev/null 2>&1 &") ;
  }
NOTE: THEIR_DISPLAY is not necessarily the same as REMOTE_HOST or REMOTE_ADDR.
You have to ask users to supply their display (set REMOTE_HOST as default).

A JAVA terminal program will accomplish something similar for the many
users with platforms that support JAVA but not X.