You can forgo the SSH session if you can directly connect from the server running the X client application to your local computer running an X server.
Connect over SSH, and then:
This will tell the X client that it should connect to the first X server and find the first X display (
your.computer.ip. Just launch the X client application as normal, and it should connect to your X server directly, and you can then close the SSH session.
You will need to make sure that a direct connection can be made from wherever the client is running to your local system, including any port forwarding and routing rules. You might have to also configure your local X server to accept connections from the network (I can't remember if it's localhost only by default). You will also need to fully disable all security on the X server (Don't do this unless you're in a private LAN where you can trust all computers):
or learn how to properly configure and set up X authentication between clients.
In all honesty, this is exactly what happens when you do it normally over SSH - Except that the SSH tunnel is being used to forward the connection, so you don't have to deal with the port forwarding, and you don't have to deal with changing your X server config to listen on the network for incoming connections (a potential security risk), and SSH will automatically configure all of the X security for your user so that the application won't have issues connecting. Even running through SSH, the application is running in the native X session, SSH is just being used as a proxy.
You didn't list the reasons for eschewing SSH, but unless it's a hard requirement handed down by a boss that you can't change, I would strongly reconsider using it for X session forwarding. You will not gain anything in particular from a direct connection, but SSH can simplify a lot of the pain points because it was designed to do just that.