Electrical signaling in biology is typically associated with action potentials, transient spikes in membrane voltage that return to baseline. Hodgkin-Huxley and related conductance-based models of electrophysiology belong to a more general class of reaction-diffusion equations which could, in principle, support spontaneous emergence of patterns of membrane voltage which are stable in time but structured in space. Here we show theoretically and experimentally that homogeneous or nearly homogeneous tissues can undergo spontaneous spatial symmetry breaking through a purely electrophysiological mechanism, leading to formation of domains with different resting potentials separated by stable bioelectrical domain walls. Transitions from one resting potential to another can occur through long-range migration of these domain walls. We map bioelectrical domain wall motion using all-optical electrophysiology in an engineered cell line and in human induced pluripotent stem cell (iPSC)-derived myoblasts. Bioelectrical domain wall migration may occur during embryonic development and during physiological signaling processes in polarized tissues. These results demonstrate that nominally homogeneous tissues can undergo spontaneous bioelectrical symmetry breaking.