Looks like a native species, Clematis virginiana. In addition to the particular leaf shape confirming ID, this is the only clematis species I am aware of that spreads by suckering as well as seeding. It blooms in late summer and early fall.

If digging it out (and you need to get ALL the roots) didn't work, then maybe it's time to head for the herbicide. Don't spray. Cut back the vines to just a couple of inches above ground and either paint the cut stems or spritz with a squirt bottle. Use glyphosate (RoundUp) full strength.

If this technique can work on Himalayan blackberries and English ivy, it should certainly work on a clematis.

Gardengal - thanks for the feedback and advice. I don't think, getting all the roots are an option, as you can see, they are buried in rocks, and are behind the property in open space which is buried in leaves, bushes, etc. I did try to dump/spray herbicide on it before, but to no avail, but I never tried the 'paint' method. Just had to google what that even meant. If I'm clear, I cut back the stems to few inches as you stated, and then literally 'paint' the cut portion of the stem using a paintbrush and herbicide (guess I'll try roundup concentrate as you suggest)? If I miss a few stems, will this method work? Thanks.

My wife just confirmed that it's for sure a clematis, but doesn't remember the species. In looking at the photos of the Clematis virginiana; it may very well be. We do get the 'seed pods' later in the year.

It may also be seedlings that you are observing as well as suckers, so you will need to be diligent. Do the cut and paint method, and check often and repeat as needed this season. You will need to check for the next several seasons as well since you had seeds. Sort of train your eyes to the leaf shape so that you spot any that sprout. I have found that 3 years or so of checking for rampant seeders will usually get rid of them, though I keep an eye out for longer. The trick is not letting it get growing well again. If you only do it once and it resprouts unattended and sets seed again, you will extend the problem.

Ok, great, I appreciate the advice. I'll try the cut and paint method and see what happens.

I had a clematis thug also it even grew up inside of our garage. On advice from an Aussie grower I waited until fall, when everything was going into the roots, then sprayed with blackberry brush killer. That was the only thing that worked I had tried digging, then the hatefull roundup, but it did not last long.