We have studied the effect of optical activity (OA) in optical fibers with multihelical distribution of refractive index profiles near the resonance wavelength, at which the conversion of an incident Gaussian beam into an optical vortex (and vice versa) takes place. We have shown that at such a wavelength the polarization vector of the input Gaussian beam rotates within the fiber at an average rate proportional to the difference in propagation constants of left- and right-circularly polarized optical vortices with the same topological charge. We also show that for certain fiber lengths the magnitude of OA can greatly exceed its average level and reach anomalously high values.