Since this is a screen it works with any projector. Depending on your need, budget and technical requirements you could consider a mini Benq. It does not have the rear projection capability but has a pretty short through. If you aim for a rear inexpensive projector, you could choose an Epson from Best Buy. It does not depend so much on the brand, whatever brand you choose you need to consider the lumens and the image ratio, and the speakers. For outdoor you probably will need an extra pair of speakers. Have in consideration that distance where you plan to place the projector and the screen since it could be that the image will be too big and ending going out of the screen.
For more information please check out our guide setting up a great backyard setup
Or take a look at our recommended projectors and screens