Keeping tomato plants alive through the fall?

My two main questions: Is it possible to keep my summer tomato plants alive and still producing tomatoes through the fall? If so, how do I do that?

I have some tomato plants that I’ve been neglecting for a while, and funnily enough they are producing more tomatoes than they ever did while I was babying them in the spring and early summer. They’ve grown out of control as well. I’m talking sprawling, twisting branches that outgrew my largest tomato cages weeks ago. Since they seem determined to live and I’m still getting tomatoes out of them, I want to take a shot at keeping them alive through the fall.

How do I clean up my tomato plants and give them a chance at surviving the fall? Any advice is welcome!