  We demonstrate that topological Dirac semimetals, which possess two Dirac
nodes, separated in momentum space along a rotation axis and protected by
rotational symmetry, exhibit an additional quantum anomaly, distinct from the
chiral anomaly. This anomaly, which we call the $ \mathbb Z_2$ anomaly, is a
consequence of the fact that the Dirac nodes in topological Dirac semimetals
carry a $\mathbb Z_2$ topological charge. The $\mathbb Z_2$ anomaly refers to
nonconservation of this charge in the presence of external fields due to
quantum effects and has observable consequences due to its interplay with the
chiral anomaly. We discuss possible implications of this for the interpretation
of magnetotransport experiments on topological Dirac semimetals. We also
provide a possible explanation for the magnetic field dependent angular
narrowing of the negative longitudinal magnetoresistance, observed in a recent
experiment on Na$_3$Bi.
