  We present the expected X-ray polarization signal resulting from distant
reprocessing material around black holes. Using a central isotropic power-law
emission at the center of the simulated model, we add distant equatorial and
axially symmetric media that are covering the central accreting sources. We
include partial ionization and partial transparency effects, and the impact of
various polarization and steepness of the primary radiation spectrum. The
results are obtained with the Monte Carlo code STOKES that considers both line
and continuum processes and computes the effects of scattering and absorption
inside static homogenous wedge-shaped and elliptical toroidal structures,
varying in relative size, composition and distance to the source. We provide
first order estimates for parsec-scale reprocessing in Compton-thin and
Compton-thick active galactic nuclei, as well as winds around accreting
stellar-mass compact objects, for observer's inclinations above and below the
grazing angle. The resulting reprocessed polarization can reach tens of % with
either parallel or perpendicular orientation with respect to the axis of
symmetry, depending on subtle details of the geometry, density and ionization
structure. We also show how principal parameters constrained from X-ray
spectroscopy or polarimetry in other wavelengths can lift the shown
degeneracies in X-ray polarization. We provide an application example of the
broad modelling discussion by revisiting the recent IXPE 2-8 keV X-ray
polarimetric observation of the accreting stellar-mass black hole in Cygnus X-3
from the perspective of partial transparency and ionization of the obscuring
outflows.
