I'm curious why you would want multiple connections to the same host in the first place. Why not use a terminal multiplexer?
ssh server remotecommand | localcommand
* set up a machine with all packages installed on some (debian/ubuntu) server: ssh server dpkg --get-selections| cut -f1 | xargs sudo apt-get install
* compare output of commands on two machines diff <(command) <(ssh server command)
* test/configure software on a local machine, plugging in its input from the roduction machine where it will be installed once ready
* avoid installing software on the server entirely; pull in the remote output and wrangle it locally (obv. not so good for high volumes)
* think a bug can be fixed by upgrading some software? try it on a test machine, pipe in the required input, and see