  Faraday rotation has been seen at millimeter wavelengths in several low
luminosity active galactic nuclei, including Event Horizon Telescope (EHT)
targets M87* and Sgr A*. The observed rotation measure (RM) probes the density,
magnetic field, and temperature of material integrated along the line of sight.
To better understand how accretion disc conditions are reflected in the RM, we
perform polarized radiative transfer calculations using a set of general
relativistic magneto-hydrodynamic (GRMHD) simulations appropriate for M87*. We
find that in spatially resolved millimetre wavelength images on event horizon
scales, the RM can vary by orders of magnitude and even flip sign. The
observational consequences of this spatial structure include significant
time-variability, sign-flips, and non-$\lambda^2$ evolution of the polarization
plane. For some models, we find that internal rotation measure can cause
significant bandwidth depolarization even across the relatively narrow
fractional bandwidths observed by the EHT. We decompose the linearly polarized
emission in these models based on their RM and find that emission in front of
the mid-plane can exhibit orders of magnitude less Faraday rotation than
emission originating from behind the mid-plane or within the photon ring. We
confirm that the spatially unresolved (i.e., image integrated) RM is a poor
predictor of the accretion rate, with substantial scatter stemming from time
variability and inclination effects. Models can be constrained with repeated
observations to characterise time variability and the degree of non-$\lambda^2$
evolution of the polarization plane.
