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When I’ve heard about tmux for the first time some months ago, first thing that came into my mind was to leave screen with its -dmS arguments away and even set tmux as yet another system shell for a couple of users. Very handy for starting up servers by scripts and having their interactive console available later without messing with pipes and log files up. Well with the simple screen at least everything was running smoothly for several years. I could not find it as easy with tmux as with screen especially having tmux set as a system shell almost from the beginning. :P now I am the fan of both tools to be used according to users preference and specific situation.
Terminal multiplexer is obviously more advanced and has wider possibilities than screen, e.g. combining the -f argument of tail in one pane with ii from suckless.org in another and yet another for orchestrating the whole leads to projects like iiii from https://github.com/tb01110100/iiii. Notice however higher resources consumption by tmux.
I imagine other users sessions take-overs by a random BOFH is slightly easier with tmux than with screen only because of pseudo-terminal permissions to be altered in the second case and client-server design of tmux in the first, but I have never tested that yet. xD
I’ve recently finished a book on tmux, The Tao of tmux. It’s available to read for free on the web. You can also grab it on Amazon as well as DRM-free in PDF/mobi/epub from leanpub.
The reason I wrote the book is I prefer the approach of separate tmux into its objects: Server, Sessions, Windows and Panes. Wrapping your brain around this concept makes everything easier since commands, config and so on will all mention them.