When the toilet is installed and flushed the water backs up in toilet and into the tub.
When the toilet is removed the water goes down just fine, and we have not noticed any of the sinks in the house causing back up in the tub.

We have removed the toilet and snaked the drain, replaced and still the same issue.

Any ideas?

Have you snaked the Vent from the roof yet? Bet its blocked.

Didn't snake from the roof, but snaked from the bottom of the vent pipe.

Didn't snake from the roof, but snaked from the bottom of the vent pipe.

Try from the top too, maybe they just didn't snake far enough, If you have to pay a plumber but have a lot of garden hose... and one of the small rotate to operate spray head hook it up and go on the roof turn the water on and lower the hose down the vent... Its worth a shot.