  The goal of this work is to characterize the polarization effects of the VLTI
and GRAVITY. This is needed to calibrate polarimetric observations with GRAVITY
for instrumental effects and to understand the systematic error introduced to
the astrometry due to birefringence when observing targets with a significant
intrinsic polarization. By combining a model of the VLTI light path and its
mirrors and dedicated experimental data, we construct a full polarization model
of the VLTI UTs and the GRAVITY instrument. We first characterize all
telescopes together to construct a UT calibration model for polarized targets.
We then expand the model to include the differential birefringence. With this,
we can constrain the systematic errors for highly polarized targets. Together
with this paper, we publish a standalone Python package to calibrate the
instrumental effects on polarimetric observations. This enables the community
to use GRAVITY to observe targets in a polarimetric observing mode. We
demonstrate the calibration model with the galactic center star IRS 16C. For
this source, we can constrain the polarization degree to within 0.4 % and the
polarization angle within 5 deg while being consistent with the literature.
Furthermore, we show that there is no significant contrast loss, even if the
science and fringe-tracker targets have significantly different polarization,
and we determine that the phase error in such an observation is smaller than 1
deg, corresponding to an astrometric error of 10 {\mu}as. With this work, we
enable the use of the polarimetric mode with GRAVITY/UTs and outline the steps
necessary to observe and calibrate polarized targets. We demonstrate that it is
possible to measure the intrinsic polarization of astrophysical sources with
high precision and that polarization effects do not limit astrometric
observations of polarized targets.
