I would consider Clematis armandii. This is an evergreen clematis that is much less cold hardy than the large flowered hybrids - certainly a good choice for warmer climates - and is especially tolerant of shady locations. In fact, it pretty much performs best in mostly shade. Armandii is an early spring bloomer and quite fragrant as well.

Most of the LFH's really need some actual sun to bloom as well as they can.

You mention that most Clematis need some sun to perform. The question I have is, if the plant is in full shade, WILL THEY NOT BLOOM AT ALL?

I am not particularly looking for exemplary or showcase award-winning blooming performance. If indeed I can get some flowering, then what recommendations are there for the best? If I am stuck with "Armandii" with the small, white flowers (boring), I can think of a dozen other plants that would outperform the Clematis any day.

When I do a search at Clematis on the Web with "shade" in the aspect box, there is a pretty long list that comes up. I can't get the actual search results to come up on this post, but you can just select "shade" for the aspect and hit the "search now" button to see the list.

You mention that your highs get to around 110, so my guess is that something from the viticella group might grow best for you as they are supposed to be quite heat tolerant. Some of the herbacious group and the jackmanii group may also do okay in the heat. Just remember that clematis will need lots of water in the heat.

Also, if you can move the trellis or support a few feet out from the north wall, that would probably allow you to have a bit of sun, at least in the summer. Unless the wall is very tall, most north side walls do get some sun in the summer. I have success with Betty Corning, Galore, Walenburg and Little Nell on a north facing wall, but the trellises are about 10 feet from the wall as they are screening my air conditioners.

Here is a link that might be useful: search at Clematis on the Web

Here are a couple of old posts discussing clematis that will grow in shade.

Old post : Any shade tolerant Clematis?
http://forums.gardenweb.com/forums/load/clematis/msg031351225998.html

old post : Need good Clem for very shaded area
http://forums.gardenweb.com/forums/load/clematis/msg0519055226820.html

Here is a link that might be useful: old post : Clematis for shade?

The recommendation is for a minimum of 4 hours of sunlight per day for proper flowering. I can't tell you your plants will never flower in your situation - only that flowering will be seriously reduced.

Shade 'tolerant' is one thing - no direct sunlight at all is entirely different. Have you considered a vine other than clematis for this area?