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We consider effects of spatial dispersion in noncentrosymmetric time-reversal invariant Weyl metals in the presence of a static magnetic field. In particular, we study currents that are linear in both the spatial derivatives of an applied electric field and the static magnetic field, which are responsible for the phenomenon of gyrotropic birefringence. We show that the chiral anomaly and the chiral magnetic effect make the leading contribution to this class of phenomena in metals. We apply the obtained results to the problem of electromagnetic wave transmission through a thin slab of a Weyl semimetal and show that the transmission coefficient contains a component that is odd in the applied static magnetic field. As such, it can be easily distinguished from conventional Ohmic magnetotransport effects, which are quadratic in the applied magnetic field. The relative magnitude of the effect can reach a few percent in Weyl materials subject to magnetic fields of 0.1 T, while the effect is several orders of magnitude smaller in metals without Berry monopoles. We conclude that the nonreciprocal optical and magnetotransport effects can be a robust probe of band topology in metals.