  The time dependence of switching multiferroic domains in MnWO$_4$ has been
studied by time-resolved polarized neutron diffraction. Inverting an external
electric field inverts the chiral magnetic component within rise times ranging
between a few and some tens of milliseconds in perfect agreement with
macroscopic techniques. There is no evidence for any faster process in the
inversion of the chiral magnetic structure. The time dependence is well
described by a temperature-dependent rise time suggesting a well-defined
process of domain reversion. As expected, the rise times decrease when heating
towards the upper boundary of the ferroelectric phase. However, switching also
becomes faster upon cooling towards the lower boundary, which is associated
with a first-order phase transition.
