  Quasi-stellar objects (QSOs) are a basis for an absolute reference system for
astrometric studies. There is a need for creating such system behind nearby
galaxies, to facilitate the measuring of the proper motions of these galaxies.
However, the foreground contamination from the galaxies themselves is a problem
for the QSO identification. We search for new QSOs behind both Magellanic
Clouds, the Magellanic Bridge, and the Magellanic Stream. We identify QSO
candidates with a combination of near-infrared colors and variability criteria
from the public ESO Visual and Infrared Survey Telescope for Astronomy (VISTA)
Magellanic Clouds (VMC) survey. We confirm their nature from broad emission
lines with low-resolution optical spectroscopy. We confirmed the QSO nature of
136 objects. They are distributed as follows: 12 behind the LMC, 37 behind the
SMC, 63 behind the Bridge, and 24 behind the Stream. The QSOs span a redshift
range from z~0.1 to z~2.9. A comparison of our quasar selection with the Quaia
quasar catalog, based on Gaia low-resolution spectra, yields a selection and
confirmation success rate of 6-19%, depending on whether the quality of the
photometry, the magnitude ranges and the colors are considered. Our candidate
list is rather incomplete, but the objects in it are likely to be confirmed as
quasars with ~90% probability. Finally, we report a list of 3609 objects across
the entire VMC survey that match our color and variability selection criteria;
only 1249 of them have Gaia counterparts. Our combined infrared color and
variability criteria for QSO selection prove to be efficient - ~90% of the
observed candidates are bona fide QSOs and allow to generate a list of new
high-probability quasar candidates.
