  We report the localization of the X-ray emission from two strongly lensed
AGN, CLASS B0712+472 ($z=1.34$) and CLASS B1608+656 ($z=1.394$). We obtain
milliarcsecond X-ray astrometry by developing a novel method that combines
parametric lens modelling with a Bayesian analysis. We spatially locate the
X-ray sources in CLASS B0712+472 and CLASS B1608+656 within 11 mas and 9 mas
from the radio source, respectively. For CLASS B0712+472, we find that the
X-ray emission is co-spatial with the radio and optical emission. While, in
CLASS B1608+656, the X-ray emission is co-spatial with radio, but displaced
with respect to the optical emission at 1$\sigma$ level, which positions this
source as an offset AGN candidate. This high astrometric precision improves on
the limitations of existing X-ray instruments by two orders of magnitude. The
demonstrated method opens a path to search for offset and binary AGN at $z>1$,
and to directly test supermassive black hole formation models in a redshift
range that has been mostly underconstrained to date.
