Testing ssh with local network work but externally will ask for password.

I have tested Gitlab internally and everything is working. Ldap, push and pull, etc. I have set up a domain and connect to Gitlab just fine and working using https://gitlab.mydomain.com. The problem I am running into is cloning, pushing, and pulling using the domain name. example: Existing Git repository
cd existing_repo
git remote add origin git@gitlab.mydomain.com:username/my-project.git
git push -u origin --all
Once it does the push, it requested me to enter the password, but if I do git remote add origin git@local IP of Gitlab it doesn't require me to enter the password because the ssh key authentication is working. I have tested with the command ssh -T git@mylocalIp and it welcome me but if I do ssh -T git@gitlab.mydomain.com it would prompt me for the password.

Start by comparing the two reported entry points, ssh -T git@mylocalIp and ssh -T git@gitlab.mydomain.com, then reproduce the push using the domain-based remote. Trace why the domain endpoint requests a password despite the local IP accepting the SSH key; done means domain-based clone, push, and pull authenticate without that prompt.