Well, what does it mean to be "grounded". There isn't something special about the voltage potential of Earth.

If a Faraday cage blocks interstellar signals only if one part of it is stuck in a ball of mud and rock... well, I have some questions.

There is the possibility of the ground being a return path to the transmitter, but if that were effective, radio infrastructure would interfere world-wide, and you could transmit through the earth's core. And even that argument would suggest that the Faraday cage should be floating, not grounded.