gevent is too much magic. It aims to give you async without changing any of your code. To accomplish this, it monkey-patches the entire Python standard library, in a way that is 99% compatible with Python, but the 1% will constantly surprise and infuriate you. Its compatibility shows no signs of increasing given how much its development has slowed down.
You can use gevent as a quick hack, but you will hate yourself if you have to maintain gevent code.