  We investigate Klein tunneling through finite potential barriers with
space-time resolved solutions to relativistic quantum field equations. We find
that no particle actually tunnels through a finite supercritical barrier, even
in the case of resonant tunneling. The transmission is instead mediated by
modulations in pair production rates, at each edge of the barrier, caused by
the incoming electron. We further examine the effect of the barrier's width on
the numbers of produced pairs in the fermionic case (characterized by
saturation) and in the bosonic case (characterized by exponential
superradiance). This work paves the way to precise studies of the radiating
dynamics of supercritical barriers, and could be applied to certain analogs of
Klein tunneling observed in systems modeled by relativistic wave equations.
