[addaon]

Probably? Transparent -- ITO on glass is the usual answer. You can deposit the ITO in patterns rather than as a continuous layer; and a conductive layer works as a faraday cage as long as the gaps are significantly smaller than the relevant wavelength (so ~mm for mmWave). So a ~500 µm grid could be laid down on glass in front of the screen, and conductively joined to a continuously conductive layer surrounding the back of the phone. The question, then, is whether the change in capacitance from a finger is observable by the touch screen through such a mesh... my intuition is that it would be, but would have to either model it or test it to find out. (Could test with just a stainless steel mesh from the hardware store.)

But none of this helps with the "toggle-able" part of the requirements…

[taneq]

I’d imagine you could have a grid of conductors which can be temporarily connected together (by a grid of transistors somehow?) forming a cage when connected. You’d only need this mechanism on the back of the case - note that a phone can still make calls when face down on a metal sheet.