  We derive the universal real time $U(1)$ topological gauge field action for
mixed quantum states of weakly correlated fermions in all dimensions, and
demonstrate its independence of the underlying equilibrium or non-equilibrium
nature of dynamics stabilizing the state. The key prerequisites are charge
quantization and charge conservation. The gauge action encodes non-quantized
linear responses as expected for mixed states, but also quantized non-linear
responses, associated to mixed state topology and accessible in experiment. Our
construction furthermore demonstrates how the physical pictures of anomaly
inflow and bulk-boundary correspondence extend to non-equilibrium systems.
