So I foudn the problem.
It’s about midway underneath the car, there is a point just after the cat. where it bolts together with the rest of the exhause, cat-back style.
The flange rotted and the flange itself broke apart. The pipe itself is in great condition.
It seems as if the heat from the cat. is almost more or less welding that pipe back together after a short bit of driving, as the pipe kind of sits inside the other part.
Now I have to see what my options for repair are. I am kind of wondering if an exhaust shop would cut out the flange, and insert a solid section. But one concern of mine with doing that is there are these intersting spring type things on the flange bolts, and there is no flex-pipe anywhere in the system.