  Multi-messenger observations of coalescing binary neutron stars (BNSs) are a
direct probe of the expansion history of the universe and carry the potential
to shed light on the disparity between low- and high-redshift measurements of
the Hubble constant $H_0$. To measure the value of $H_0$ with such observations
requires pristine inference of the luminosity distance and the true source
redshift with minimal impact from systematics. In this analysis, we carry out
joint inference on mock gravitational wave (GW) signals and their
electromagnetic (EM) afterglows from BNS coalescences and find that the
inclination angle inferred from the afterglow light curve and apparent
superluminal motion can be precise, but need not be accurate and is subject to
systematic uncertainty that could be as large as $1.5\sigma$. This produces a
disparity between the EM and GW inferred inclination angles, which if not
carefully treated when combining observations can bias the inferred value of
$H_0$. We also find that already small misalignments of $3^{\circ}-6^{\circ}$
between the inherent system inclinations for the GW and EM emission can bias
the inference by $\mathcal{O}(1-2\sigma)$ if not taken into account. As
multi-messenger BNS observations are rare, we must make the most out of a small
number of events and harness the increased precision, while avoiding reduced
accuracy. We demonstrate how to mitigate these potential sources of bias by
jointly inferring the mismatch between the GW- and EM-based inclination angles
and $H_0$.
