Ok, just in case anyone wants a cloudy plastic for Eevee, I’ll stick what I ended up with here:
Weirdly, to get rid of the internal reflection (that even shows up in Cycles as mentioned above), I had to remove all transparency and only use transmission… kinda counter-intuitive, like with most things Blender (and I basically stumbled on the solution by accident after reading the Blender secrets e-book). 