Ask Mr. Smarty Plants

Finding a vine that will fit your requirements is quite a challenge, especially with full Arizona sun!  Unfortunately, most plants flower at some point in their lives.  However, some plants produce more or larger flowers than others, so the next best solution is probably to plant something that has small, sparsely produced flowers.  A good match for your preferences is the Vitis arizonica (Arizona Grape).  This vine will grow in full sun, is very hardy, and is native to Arizona, so should tolerate its harsh summers very well.  When it does produce flowers (young grape vines usually don’t produce any flowers in the first couple years) they are small and will probably not make a mess of your pool.  As the vine gets older it will start to produce flowers that will turn into purple fruit, so that is something to consider.  Though, birds and other wildlife are likely to consume the fruit before it makes a huge mess.

Here are a couple more options that should tolerate full sun and should be relatively clean plants:

Macroptilium sp. (Purple bushbean) is another option that would do well full sun and has small flowers. 

Passiflora foetida (Corona de cristo) could also work well for you.  It is a very hardy vine, which is happy in full sun, and the flowers are large, but normally only sparsely cover the vine.    

To find any of these plants its best to contact any local native plant nurseries and inquire if they have the species.  The Desert Botanical Gardens spring plant sale is also coming up in March and the Passion Flower, Arizona Grape, and Purple Bushbean are listed as plants they will be selling.