We have Clematis climbing up through Jasmine. Problem is that the Clematis needs pruning, but the Jasmine does not really. Climbing roses and Clematis are often grown together too, though not by me as I hate roses. Again the pruning regime has to be similar for ease.
We also have Clematis growing up through Ivy, not the other way round. That works as the Ivy may be pruned at any time.

yes as Owd says it can be done. The tropaeulum dies down every year so it shouldn't be a problem.

As Owdboggy and Seaburngirl say, it can be done, but it makes maintenance difficult, the faster growing climber often kills off the other one, and even if they work physically, aesthetically they will be a horrible brush pile!

I've used Mandavilla vine which is a fast growing tropical vine which is in constant bloom all growing season in combination with clematis, and honeysuckle....I even have sent it up my tall growing zebra grass(you can see this in my photo section).The great thing about it that it dies at the end of the season and the vine gets very brittle and crumbles away. This way I can change colors next season or just leave the primary vine as is.

Many thanks potential problem solved. Now to get a Tropaeulum and get it to grow.