Oddly enough I searched for a fix for this after posting. And amazingly I found one that works (as well as several that didn't, and one that require add-ons)

For anyone interested, this is in my .tmux.conf now

    unbind -T copy-mode-vi Enter
    bind-key -T copy-mode-vi Enter send -X copy-pipe "xclip -selection c -i" \; send -X clear-selection
    bind-key -T copy-mode-vi MouseDragEnd1Pane send -X copy-pipe "xclip -selection c -i" \; send -X clear-selection

You would have to switch the xclip for wl-copy if you're using Wayland, and some form of pbcopy command on Mac. No idea what windows folks do.