Dear BB:

You wrote:
> I know that it is “Dave and Alex’s project” not “Dave’s and Alex’s project” (if they are working together on one project), but if it were Alex and I working on the project is it “Alex and my project” or “Alex’s and my project”? The first seems to follow the rule, but the second sounds more correct to me.
>
Sometimes the sound is not technically correct. The first follows the rule and is the form to use in standard writing. A lot of time you may hear the second in speech. Interestingly, people seem to have less of a problem with “Alex and his project” or “Alex and her project.”

A lot of times in both writing and speaking we say “the project of Alex and mine” to avoid any ambiguity.