  We investigate the impact of the Adler-Bell-Jackiw anomaly on the
nonequilibrium evolution of strong-field quantum electrodynamics (QED) using
real-time lattice gauge theory techniques. For field strengths exceeding the
Schwinger limit for pair production, we encounter a highly absorptive medium
with anomaly-induced dynamical refractive properties. In contrast to earlier
expectations based on equilibrium properties, where net anomalous effects
vanish because of the trivial vacuum structure, we find that out-of-equilibrium
conditions can have dramatic consequences for the presence of quantum currents
with distinctive macroscopic signatures. We observe an intriguing tracking
behavior, where the system spends longest times near collinear field
configurations with maximum anomalous current. Apart from the potential
relevance of our findings for future laser experiments, similar phenomena
related to the chiral magnetic effect are expected to play an important role
for strong QED fields during initial stages of heavy-ion collision experiments.
