You are in the position of a Linux user with a multi-monitor setup who wants to run TF2, but you keep encountering an issue while running TF2 when alt-tabbing out and back in the game. The issue usually manifests it's self in either one of these ways:

• When you alt-tab out of the game and back in, TF2 only takes up a portion of your monitor space. This may be accompanied with various graphical glitches
  
• When you alt-tab out of the game and back in, TF2 appears on the wrong monitor

You may have tried googling this issue to see what might be causing it and a fix, but all you find are bug reports of the issue you are encountering or irrelevant links, but nothing actually detailing how to fix the issue and be able to play TF2 on a multi-monitor setup

There is a very simple fix to this issue.

Step #1: Go to TF2 in your library, right-click and click properties
Step #2: Go to launch options and add -windowed
What you are going to do next depends on whether you'd like the game to be in a window or borderless fullscreen
Step #3a: If you'd like the game to be in borderless fullscreen, add the following to launch options
Step #3b: If you'd like the game to be in a window, determine how many pixels your panel(s) and window titlebar take up and subtract that from the resolution of your main monitor. In my case I have a 2560x1440 monitor and the top panel + titlebar take up 63 vertical pixels leading me to launch the game at 2560x1377. If you do not perform this step while running in windowed, TF2 will be cut off a bit at the bottom.
Step #4: Close the dialog, go to TF2 and launch the game

Congrats! You can now play TF2 and alt-tab out of it without that glitch occurring.

This section is currently incomplete and will likely be updated in the future.

Problem #1: a panel is in the way or is still visible in borderless fullscreen

Fix: This largely depends on your WM and DE. You'd be better off googling the answer to this questiion depending on what WM/DE you're running or what distribution you are using provided you haven't made any changes to the WM/DE. If you still can't find anything or not willing to solve the problem it might be best to run TF2 in Windowed mode.