ZDoom

This already exists. Set 'vid_activeinbackground' to 1 in the console and it should do what you want. For technical reasons this only works in windowed mode, not in fullscreen.

Currently when gzdoom window is not in focus, the game will pause.
However, I think, that allowing the game to run (and intercept controller input) while the window is not focused akin to source games would allow for a makeshift split-screen experience by connecting to a localhost server.
Here is a representation of how that would look: https://imgur.com/4Br9yPX
Something like this https://youtu.be/LutfKPr_hIc?t=242 except without the use of third party virtualization utilities.