  We study Josephson junctions based on inversion-asymmetric but time-reversal
symmetric Weyl semimetals under the influence of Zeeman fields. We find that,
due to distinct spin textures, the Weyl nodes of opposite chirality respond
differently to an external magnetic field. Remarkably, a Zeeman field
perpendicular to the junction direction results in a phase shift of opposite
sign in the current-phase relations of opposite chirality. This leads to a
finite chirality Josephson current (CJC) even in the absence of a phase
difference across the junction. This feature could allow for applications in
chiralitytronics. In the long junction and zero temperature limit, the CJC
embodies a novel quantum anomaly of Goldstone bosons at $\pi$ phase difference
which is associated with a $\mathbb{Z}_2$ symmetry at low energies. It can be
detected experimentally via an anomalous Fraunhofer pattern.
