Here’s what I’d suggest doing: Next time you have the problem (or next time you can induce the problem) go ahead and let the monitor go to sleep. Restart, then open Eve and immediately file an in-game bug report. We might be able to see the logging from your previous session with your report, but even if not, the hardware and software rundown of your machine will be super-helpful in tracking this down.
Also, it’s possible that different power settings or even a different monitor cable may solve this. If you search “monitor sleeps randomly” on Google, you’ll find various sites with instructions on how to adjust the power settings. I’m hesitant to link one directly because I’m not sure which advice is most accurate or comprehensive. (Even if you’re only seeing this problem in Eve, it could be something like an interaction between your power settings and what Eve is doing, so this might be a useful workaround.)