Well i'm not sure if this will help but the SSL default port is 443, did you try http://mydomain.com:443/dir/perl.pl?
UPDATE: I lied. that's the SSH default port. Downvote as needed :-(
UPDATE2: Just so no one gets screwed up, The Mad Hatter is correct. Pay no mind to the above babble about ports.
http://mydomain.com:443 would just address a http server running on port 443 since to protocol is http; vice versa, a https server running on port 80 would be called as https://mydomain.com:80