Common name: Sweet Autumn Clematis
Family: Ranunculaceae
Genus: Clematis
Species terniflora
Plant Link: http://plantsdatabase.com/go/683/
Neato! What a great place to showcase this plant! I have tried to grow it for 2 years and it just doesn't grow much and absolutely no flowers! Must be my brown thumb because the photos posted on Dave's sure tell a different story! I like your greenhouse setting too! Thanks for posting the picture.
Sorry to hear that adairia. Want me to send you some of my bugs? lol. All I can recommend is repeated drastic prunings to control size and keep it from setting seed. I have not tried herbicides, but the plant is so woody it would probably need several doses. If you use Roundup, be sure to spray on a sunny day, using a fine mist, so that the plant can take up the chemical through it's tiny stoma on the leaf undersides. Good luck!
Toxico, with a roll of nylon fishing line, some S-hooks, and a partner to play pitch and catch over the greenhouse, you could give the vine a little help covering the whole greenhouse roof.