[DiscoDuck (Paul)]

In PixInsight, you can register them to the union of all the images I believe with Register Union Separate as the working mode in the StarAlignment process. You'd get each image then with a black border. Then you could stack these and crop the result as you desire.
(The pixel rejection algorithms may freak out at multiple black images at some pixels and reject the true signal, but that's ok, as you're only interested in the intersection of your images).

Edit: Just had a look and it may be messy to do it like this ... at least more than my first guess .... as I don't think you can do all images at once but have to kind of add them one by one and keep updating the reference image to the new one until all accumulated?? Not sure. But sort of create a mosaic from them all, then register the individual images against that mosaic and then stack them. Hmm. Not easy. Perhaps a PI wizard has a slick way to do it.

Or alternatively, run StarAlignment in the normal mode with a randomly chosen reference image, take the average of the computed offsets for each image to get the center of mass, and pick as a reference image the frame closest to that average offset. Then run again with that as the reference.