It sounds romantic, but the reality is none will work well. Build a trellis or pergola.

Catsclaw or Lady Banks Rose - totally overgrows and kills the tree with the dense shade of the vine. Weight of live and dead parts can break the branches.

Queens wreath - very dense, tends to make a "hat" on top of a tree.

Bougainvillea - add thorns all the problems with the others.

When the vine freezes you have a tree full of dead vines and a real PITA cleanup job. Ripping the dead vines down can damage the tree. Build up of dead material provides nesting places for tree rats and roaches.

hmm. one of my books really likes adding vines to trees, and has a list of suitable smaller vines which won't overwhelm, and which are native or suited to hot climates. I'll take a look if I remember and give you the list.

Okay, this is from one of my favorite desert books, The Hot Garden by Scott Calhoun. I make no representation about this from personal knowledge.

"[The deserts] have a great wealth of vining plants--an abundance of garden designer drapery. For garden design purposes, vines really fall into two categories: large vints for screening and shade, and smaller vines for layering, texturing, and jewel-like flowers. The petite vines are best kept used rambling up cholla skeletons, ocotillos, yccas, agaves, or small trees. . . "

There's a list of small and large vines either native or hardy adapted for the desert, if you're still interested.