We present a general model for the spatial coherence function of random fields created by scattering elliptical, perfect vortex beams. Remarkably, as opposed to the free-space propagation of typical random fields, there are regimes where the spatial coherence function does not evolve. We demonstrate analytically, numerically, and experimentally that both the size and the shape of spatial correlations can be precisely controlled in a manner that is independent of propagation distance.