I usually connect to a single server via PuTTY, and have three (horizontally) split sessions for logs, and another PuTTY window for sysadmin, and maybe one for coding (with multiple sessions via byobu), and possibly another one for reference. This doesn’t seem to be possible with tmux as backend for byobu.

This may be because, tmux, in spite of all the praises it receives for speed, features, and whatnot, doesn’t support multiple concurrent, but distinct sessions via byobu. I haven’t researched this deep enough to conclusively say that there is absolutely no way to do this, but I’ve found simply switching over to screen to be much easier solution for now.