An active conducting medium is one where the resistance (conductance) of the medium is modified by
the current (flow) and in turn modifies the flow, so that the classical linear laws relating current and
resistance, e.g., Ohm’s law or Darcy’s law, are modified over time as the system itself evolves. We consider
a minimal model for this feedback coupling in terms of two parameters that characterize the way in which
addition or removal of matter follows a simple local (or nonlocal) feedback rule corresponding to either
flow-seeking or flow-avoiding behavior. Using numerical simulations and a continuum mean field theory,
we show that flow-avoiding feedback causes an initially uniform system to become strongly heterogeneous
via a tunneling (channel-building) phase separation; flow-seeking feedback leads to an immuring (wallbuilding) phase separation. Our results provide a qualitative explanation for the patterning of active
conducting media in natural systems, while suggesting ways to realize complex architectures using simple
rules in engineered systems.