  The thermodynamic uncertainty relation gives a lower bound on the amount of
dissipation in a mesoscopic system. By considering the fluctuations in the
hysteresis of the current -- the sum of the currents in the time-forward and
time-reversed processes -- we extend this relation to systems with broken
time-reversal symmetry, either due to the presence of odd state variables, odd
driving field or due to explicit time-dependent driving that is time-reversal
asymmetric. We illustrate our predictions on a dilute, weakly-interacting gas
driven out of equilibrium by the slow compression of a piston and on a
ballistic multi-terminal conductor with an external magnetic field.
