  A significant contribution to the electromagnetic radiation by a fast
electric charge moving in anisotropic chiral matter arises from spontaneous
photon radiation due to the chiral anomaly. While such a process, also known as
the "vacuum Cherenkov radiation", is forbidden in the QED vacuum, it can occur
in chiral matter, where it is more appropriate to call it the "chiral Cherenkov
radiation". Its contribution to the radiation spectrum is of order $\alpha^2$
compared to $\alpha^3$ of the bremsstrahlung. I derive the frequency spectrum
and the angular distribution of this radiation in the high energy limit. The
quantum effects due to the hard photon emission and the fermion mass are taken
into account. The obtained spectra are analyzed in the case the quark-gluon
plasma and a Weyl semimetal.
