Hello,
If I recall correctly, you need to modify the configuration.php file to reflect the FQDN - the actual URL vistors use to reach the site.
I believe that this is the line:
mosConfig_live_site = 'http://www.domainname.com';where
domainname would be your site's domain name.
After you have modified this file you will find that if internal users try to view the site using an IP address or something other non FQDN they will see the site without the formatting (just as your external visitors see it now).
This is from memory, but check the Joomla site, you should find the exact answer correctly there.