My local tmux and one remote tmux both used Ctrl-k as their prefix. After connecting over SSH, Ctrl-k c still created a local window. The local tmux consumed the prefix before SSH could send it to the server.

I wanted Ctrl-k to control the remote when its tmux also accepted Ctrl-k, but stay local for other servers. The working solution uses a pane-local flag and one SSH probe per login. Ctrl-b remains an always-local escape hatch.