Pull request containing the same changes as #1258, created since #1258 does not allow edits from maintainers and is currently out of date with master, and therefore cannot be merged even though the PR is approved.

From @guiscaranse (original author on #1258):

This patches the eventlet library to allow it to run without issues under the Sentry SDK when using greenlet>=0.5. Eventlet doesn't pin the greenlet dependency and eventlet/eventlet#663 (that's why the sdk runs nicely under Gevent for example). So, the eventlet library itself is not a good marker to whatever the contextvars are broken or not.

This should fix #1036