Figs do well in Arizona, but are high water users. Be sure to water deeply (to a depth of three feet), but infrequently. In the summer that means watering once every two weeks tapering to once a month in the winter. If your tree has a basin around it, be sure it extends to the edge of the canopy which is where 90 percent of the feeder roots are located.

It's hard to tell which variety you have from the photo (I could only see one image). I've listed the most popular figs for the low desert with brief descriptions.

Black Mission: Popular and flavorful for eating fresh or dried. Medium to large pear-shaped fruit has skin which is black-purple and strawberry-colored flesh. Tree is large and long lived. Well adapted to elevations below 2,000 feet.

Brown Turkey: Sweet tasting and best eaten fresh. Bell shaped medium to large fruit has browish- purple skin with pink flesh. Tree is large and best adapted to 2,000 - 3,000 feet. Not as prolific as Mission.

Conadria (White): Mild and sweet with whitish-pink flesh. Large fruit with cream to light green colored skin. Good, eaten fresh or dried. Tolerates heat well.

White Kadota: Good, fresh or dried, suitable for canning. Fruit is medium sized with yellow skin. Flesh is amber with few seeds. Hot weather aids in ripening

I have kodata and the leaf shape and overall tree structure isn't quite right for kodata. Not sure what it is though.
RE the taste - figs commonly produce 3 crops a year. the first, called the breba crop, usually is only a few figs but large and good quality. the second will be the smallest figs and the most of them. The third on some varieties is good but on many it never really ripens correctly. Sounds like you were tasting the third crop of the year. Since your third crop did not ripen that probably rules out Brown Turkey.
The new crop should be getting rather nice about now. If it is a green fig, it will just turn less deep green to a light green or pale yellow. Other figs will turn brown or purple.
Another possibility for green fig may be Genoa
Another thing you may want to try is to post over in the fig forum. There are some real figgy folks over there
_smiles his jolly smile_