A conservative approach since you don't know the type of clematis you have would be to prune half of the older thick stems back to within 12 inches of the ground. Prune the other half back to three feet or so. If the more severe pruned stems don't resprout, then you still have the other stems to produce more growth.

I would then make sure to fertilize the plants in early spring with a good fertilizer that is balanced or has a slightly higher second number (phosphorus content) since it stimulates root development and flower production.

Thanks for the quick reply, Nick. Any harm in cutting them all at 3 feet? There are only 3 main trunks, with myriad branching starting around the 5-foot level.

Should I cut them now? Put anything on the cuts?

Nope nothing is needed on the cut stems of clematis. If you want to try 3 feet or so, then go for it. All of my clematis were cut back in December and early January. Now should probably be fine to cut them back.

Let us see pictures when it finally blooms. Perhaps we can help with the id of it.