Electromagnetic random beams with non-uniform source correlations have been recently shown to develop, on propagation in free space, the regions in transverse cross-sections where the degree and the state of polarization can significantly differ from those beyond that region. The size of the region and the values of polarimetric properties in it can be fully controlled from the source plane. In this paper the influence of a random isotropic medium on such beams is shown to suppress the effect in several ways, in particular by shifting the location of the region back to the axis.