Thanks for the nice comments
I'm somewhat reluctant to believe that the puzzle jumbles. Wouldn't these cut lines be enough?
You could unbandage the puzzle further using something close to the cut lines you've drwn, but you could not unbandage it completely without fudging. Here's an image of one face with the cut lines you've drawn:
Quadstar Partial Unbandaging.JPG [ 51.98 KiB | Viewed 1328 times ]
Note that the small lines slicing the face center do not quite line up, and the center is not a true octagon. In order to make this face center work, it would need to be made into a circle, and the tiny pieces around it left out. The eight smaller triangular pieces adjacent to the center would also need to be fudged to match the two on the left and right.
I'm pretty sure the dual of this is Oskar's PentaJumble.
Would it be possible to assemble in such a way to eliminate the jumbling/bandaging? i.e. without the reversed sections
If you reverse two of the edges you can get two complete stars, and two stars with two arms reversed. The puzzle cannot be assembled to make four complete stars at the same time.