How to enable (and use) persistent remote connection with VSCode/tmux?

Just do whatever you do normally, all methods to open a new REPL should reconnect automatically.

Alt-J Alt-K will kill the Julia process and ask you whether you want the tmux session to close as well. Otherwise you should be able to close the dead pane with Ctrl-B X Y too.

Enable mouse mode in tmux by putting set -g mouse on into your ~/.tmux.conf.