I've had some success planting them together. My strawberry roots
tend to be around 2-6" deep while the asperagus is much deeper, more like 6-12".
In my case, the strawberries performed the best out of the pairing. I didn't have as much luck with the asperagus, but that was mainly due to the presence of asparagus beetles.
Or perhaps the beetles attacked the aparagus plants because they were weakend due to lack of nutrition (??). FWIW, I'm still getting a few spears here and there after 4 years.
I'd say it's worth a shot, especially if you are short on space.