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We present the expected X-ray polarization signal resulting from distantreprocessing material around black holes. Using a central isotropic power-lawemission at the center of the simulated model, we add distant equatorial andaxially symmetric media that are covering the central accreting sources. Weinclude partial ionization and partial transparency effects, and the impact ofvarious polarization and steepness of the primary radiation spectrum. Theresults are obtained with the Monte Carlo code STOKES that considers both lineand continuum processes and computes the effects of scattering and absorptioninside static homogenous wedge-shaped and elliptical toroidal structures,varying in relative size, composition and distance to the source. We providefirst order estimates for parsec-scale reprocessing in Compton-thin andCompton-thick active galactic nuclei, as well as winds around accretingstellar-mass compact objects, for observer's inclinations above and below thegrazing angle. The resulting reprocessed polarization can reach tens of % witheither parallel or perpendicular orientation with respect to the axis ofsymmetry, depending on subtle details of the geometry, density and ionizationstructure. We also show how principal parameters constrained from X-rayspectroscopy or polarimetry in other wavelengths can lift the showndegeneracies in X-ray polarization. We provide an application example of thebroad modelling discussion by revisiting the recent IXPE 2-8 keV X-raypolarimetric observation of the accreting stellar-mass black hole in Cygnus X-3from the perspective of partial transparency and ionization of the obscuringoutflows.

X-ray polarization properties of partially ionized equatorial obscurers around accreting compact objects