You can let vim catch the mouse clicks / drags and have it manage you X11 clipboard (start looking at vim help for :set mouse=...). Personally I prefer to not have vim mess with my mouse and X11.
I think it's better to just use vim's horizontal selection ("visual line", Shift+v) and use yank (y key) or filter it into an extern command like "xlip -i -selection clipboard" to copy the highlighted text into the clipboard.
Anyway, using tmux for selection in vim is a bad idea. You can give vim mouse support by `set mouse=a`, which will prevent tmux from capturing selection and scrollback control if you like. Achieving parity between vim, system, and tmux clipboard (and for bonus points, across a network) is left as a "fun" exercise.
https://awhan.wordpress.com/2010/06/20/copy-paste-in-tmux/
I'm just learning all of this, but since it's a suggestion from some smart people I bet, just like in vim, that you can do a lot more once you've overcome the first pain of using it correctly.
set clipboard=unnamedplus
Then all yanking and deleting will put the yanked or deleted text in the X11 clipboard, from whence you can paste it in other applications, including terminals. For more information try help registers
I use this all the time, and find it essential (no mouse required).Instead I use i3 to split screens, etc. much better imo.
Still often use tiling though to tile a browser with a terminal or two browsers.