  Berry phases strongly affect the properties of crystalline materials, giving
rise to modifications of the semiclassical equations of motion that govern
wave-packet dynamics. In non-Hermitian systems, generalizations of the Berry
connection have been analyzed to characterize the topology of these systems.
While the topological classification of non-Hermitian systems is being
developed, little attention has been paid to the impact of the new geometric
phases on dynamics and transport. In this work, we derive the full set of
semiclassical equations of motion for wave-packet dynamics in a system governed
by a non-Hermitian Hamiltonian, including corrections induced by the Berry
connection. We show that non-Hermiticity is manifested in anomalous weight rate
and velocity terms that are present already in one-dimensional systems, in
marked distinction from the Hermitian case. We express the anomalous weight and
velocity in terms of the Berry connections defined in the space of left and
right eigenstates and compare the analytical results with numerical lattice
simulations. Our work specifies the conditions for observing the anomalous
contributions to the semiclassical dynamics and thereby paves the way to their
experimental detection, which should be within immediate reach in currently
available metamaterials.
