I have no idea why this happens, but on one of my machines, tmux swallows Ctrl+Tab and Ctrl+Shift+Tab. I have other machines, running the same tmux version and config, where this doesn’t happen. The only significant difference I’m aware of is that I access the other machines via SSH, but I’ve tried ssh localhost and the problem persists.

I gave up trying to work out the reason, and resorted to explicitly configuring tmux to send on the relevant escape sequences:

# Pass on Ctrl+Tab and Ctrl+Shift+Tab
bind-key -n C-Tab send-keys Escape [27\;5\;9~
bind-key -n C-S-Tab send-keys Escape [27\;6\;9~