  We investigate a symmetry protected $Z_2$ topological electron glass -- a
glassy equivalent of the $Z_2$ topological band insulator in crystalline
systems -- and uncover associated quantum phase transitions in this three
dimensional amorphous network of atoms. Through explicit numerical calculations
of the Witten effect, we show that the $Z_2$ glass is characterized by an
anomalous electromagnetic response -- dyons with 1/2 electronic charge. We
further study, using a variety of numerical diagnostics including such
electromagnetic responses, the phase transitions of the $Z_2$ glass into a
metallic and/or a trivial insulating phase. We find that the phase transitions
here are governed by subtle features of mobility edges and "spectral inversion"
which are possibly unique to structurally amorphous systems. Our results
provide a concrete setting to understand the general underpinnings of such
phases -- where strong structural disorder interplays with symmetry-protected
topological order.
