I've had great fun with my Hall's Honeysuckle. It climbs all over to the point where it snared my hanging plants, and only the hardest frost put an end to green leaves for all of a month. Plus, lovely sweet flowers that hummingbirds love. When I lived in CA we had one that covered a whole fence and stayed green all year.

I had an American Beauty honeysuckle but that lost all its leaves in the first frost and hasn't put out any more yet.. I'm starting to wonder if it's dead.

You also might try Star Jasmine; it doesn't grow as fast (altho it might, in your area) but it is more reliably evergreen and even sometimes has a second flush of blooms around Christmastime.

Jasmines and Bower Vines. Morning glories are annuals and they drop leaves and seeds everywhere. Mine aren't evergreen but they are evergrowing.

I'll second the Star Jasmine, also known as Confederate Jasmine. It follows the old adage of first year sleep, second year creep, third year leap. Once it's established though it is beautiful year round. If it can thrive in humid and HOT Houston it should love Cali.