I learned this from the former editor of the Sunset Western Garden book, Dick Dunmire. He was a very witty and knowledgeable man, and dearly missed.
If it grows upon a lattice
The hoipoloi call it cle-ma'-tis
But Dr. Webster would insist
That we call it Clem'-uh-tis
So that's how I pronounce it- Clem'-uh-tis