Avoid getting too much of the surface (looking upward) in the shot to not get this effect.
So that means trying to stay even more parallel to the surface. I'm not sure if that will be possible in practice. I was already so close to the surface, that with the camera in portrait direction, I already had to be careful not to breach the surface with the large domeport and accidentally turn it into a half-half shot.
It's one situation in which a lens with a ninety-degree field of view is better than a fisheye
If the large field of view of the fisheye lens (180 degrees) is to blame, then I should get a better result if I had used the Tokina at the 17mm end (100 degrees), right? Would it be a good idea to use a 1.4x teleconverter to reduce the field of view?
otherwise you need to control the background that is visible in Snell's window: turning off any lights and shooting with a small aperture, framing the sky outside...
That will only underexpose (darken) the sky (or in my case the roof of the swimming pool) that is visible through Snell's window. It will not make it reflective.