There are many beautiful vines that will grow and flower in our climate: some are annuals, some perennials, some lose their leaves in winter, and some remain evergreen.
Here's an idea for a nice evergreen vine to grow on a trellis, fence or lattice and it is one that has done well for me-Jasmine polyanthum. One of the common names for it is "winter jasmine," but that name is also used for several other types of true Jasmines, as well as some plants that aren't Jasmine at all. It is available in many places in the local area.
J. polyanthum is fast growing to about 20 feet tall. It will get over 10 feet tall in just a year or so from planting if it is happy. Mine grows on the North side of my house on a lattice attached to the back patio. It can take full sun, but in our hot summers I think it does better with a little afternoon shade. Prune it after flowering and just like Jack's beanstalk, it will climb right back up in no time. It is a "light weight" vine, which can climb on lattice work without getting too heavy for its support.
It's hard to decide what is the best feature of J. polyanthum, since it stays green the year around and in April (in Sierra Vista) it covers itself with clusters of blooms that are pink in bud and white upon opening. They are sweetly fragrant, and will perfume a large area such as the whole patio when planted nearby. Humming birds seem attracted by the flowers, and even though they bloom here so early (before many of the hummingbirds have returned from their winter homes) we do have some year 'round residents-mostly Black Chinned Hummingbirds. I keep two small feeders up for them in winter, and then when spring arrives and more birds return, I put up the larger feeders in addition to the small ones. This year we had a hummingbird nest in the tangle of our J. Polyanthum vine, and we could watch developments from the kitchen window.