By 'edible' I assume you mean fruit trees? They are typically not referred to as 'shade trees' as they seldom get large enough to cast much shade and their canopies need regular thinning to encourage fruit production

I don't know if you qualify as 'low desert' but here is a list provided by your extension service - always the best local resource available - of AZ suitable fruit trees. If you want shade, you may have to select something else that will grow to an appropriate size but not necessarily provide any edible fruit.

A good choice would be tamarind but a young tree will need some help as it would be killed by frost. Once established it's fairly drought resistant and can take some cold. The fruit is very tart but is prized in the tropics made into drinks and candy or used as a souring agent.

Find out if pecan trees will grow well where you live.

Contact your local State Department of Labor office to see if they can put you in contact with someone with a tractor to plow up your yard and "naturally" fertilize it, plowing under either chicken manure and other natural fertilizers. Yes, the smell of it will truly reek until the ground firms back up "seals" itself but spreading hay over the plowed ground after you add grass seed will help a bit. Doing that will vastly improve the likelihood that any grass or orchard, etc., will grow well.

What zone in Arizona? Warmer zones avocado & white sapote can get very large and are evergreen. Figs and mulberry are deciduous shading the hottest months and dropping leaves in winter if you wanted sun then. I saw large versions of all these trees when I lived in San Jose 9b zone.

Go to your local garden center they will be more helpful than all of us guessing and especially without pictures . The soil needs to be ammended no matter, even more so if you plan on trees . What is the watering restrictions where you live trees take lots of water at least when newly planted.IMO I would embrace the catus type landscaping that always does well in AZ. Get some fabulous umbrellas for shde and if the heat is coming into the house get solar shades to control the heat and the UV rays that fade farics floors and rugs.IMO grass is the last itme I would try to grow in the desert look around and see what nature grows well where you live that is the best choice for your yard.

For bad soil adding organic matter will help. Many tree trimmers will give you wood chips which you can use as mulch and/or leave in a pile to compost over time. Try chip drop.com if you are having trouble finding someone

I already linked to an AZ extension service publication that lists fruit trees suitable to grow in that climate......the extension service is much more knowledgeable about growing conditions and suitable plant selection than any nursery.

And no, the soil does not need to be amended no matter what! As long as you have decent drainage, it has long been proven that planting in the indigenous native soil is going to produce the best results. Amending - unless you amend the entire garden before planting anything - can interfere with soil interface and water percolation issues.

The OP only asked about edible or fruit bearing shade trees suitable for Arizona. She didn't ask about a lawn or a xeric desert garden. Let's not assume issues that are not yet stated or desired.