I'm currently stuck with a laptop running Windows at work, but I do all my development on a server running Linux, so I end up working in a tmux session over PuTTY all day.
I wanted to copy text between vim instances running in different windows in my tmux session. I tried using the * register to copy to the system clipboard, but since I'm connected through PuTTY I don't have an X session, so there is no system clipboard, even if I launch gVim instead of vim. (I might have been able to use x-forwarding or something to fix this, but I didn't want to install an x-server on windows.)
I thought that there should be a way to use the tmux copy/paste buffer from vim in place of the system keyboard, and sure enough someone has written a plugin for that.
Installing fakeclip adds a new register, &, which maps to tmux's paste buffer. Since all my vim instances are within the same tmux session, this makes it super simple to yank/put text between them.
The plugin documentation says that fakeclip should also work with gnu screen, but I haven't personally tested that.
tl;dr
Install the fakeclip plugin, and then you can use "&y to yank into tmux's buffer, and "&p to put from tmux's buffer.