Several people mentioned Kitty, so I looked at the screencast on the Kitty website to see what features Kitty has. While it has some neat features, like showing the current directory or the currently running command, I'm pretty sure that running tmux or GNU screen would break those features and most others.

So it looks to me like you can use a fancy terminal emulator with cool features, or you can use a terminal multiplexer (tmux or GNU screen), but the two don't mix.