  We investigate a sample of 3412 {\it International Celestial Reference Frame}
(ICRF3) extragalactic radio-loud sources with accurate positions determined by
VLBI in the S/X band, mostly active galactic nuclei (AGN) and quasars, which
are cross-matched with optical sources in the second Gaia data release (Gaia
DR2). The main goal of this study is to determine a core sample of astrometric
objects that define the mutual orientation of the two fundamental reference
frames, the Gaia (optical) and the ICRF3 (radio) frames. The distribution of
normalized offsets between the VLBI sources and their optical counterparts is
non-Rayleigh, with a deficit around the modal value and a tail extending beyond
the 3-sigma confidence level. A few filters are applied to the sample in order
to discard double cross-matches, confusion sources, and Gaia astrometric
solutions of doubtful quality. {\it Panoramic Survey Telescope and Rapid
Response System} (Pan-STARRS) and {\it Dark Energy Survey} (DES) stacked
multi-color images are used to further deselect objects that are less suitable
for precision astrometry, such as extended galaxies, double and multiple
sources, and obvious misidentifications. After this cleaning, 2642 quasars
remain, of which 20\% still have normalized offset magnitudes exceeding 3, or
99\% confidence level. We publish a list of 2118 radio-loud quasars of prime
astrometric quality. The observed dependence of binned median offset on
redshift shows the expected decline at small redshifts, but also an unexpected
rise at $z\sim 1.6$, which may be attributed to the emergence of the C IV
emission line in the Gaia's $G$ band. The Gaia DR2 parallax zero-point is found
to be color-dependent, suggesting an uncorrected instrumental calibration
effect.
