  Type II Cepheids (T2C) are less frequently used counterparts of classical
Cepheids which provide the primary calibration of the distance ladder for
measuring $H_0$ in the local Universe. In the era of the Hubble Tension, T2C
variables with the RR Lyrae stars (RRL) and the tip of the red giant branch
(TRGB) can potentially provide classical Cepheid independent calibration of the
cosmic distance ladder. Our goal is to provide an absolute calibration of the
Period-Luminosity, Period-Luminosity-Color and Period-Wesenheit
relations(PL,PLC and PW) of T2Cs in the Large Magellanic Cloud (LMC). We
exploited time-series photometry in the near-infrared (NIR) bands for a sample
of more than 320 T2Cs in the Magellanic Clouds (MC). These observations were
acquired during 2009-2018 in the context of the VMC ESO public survey (The
VISTA near-infrared YJKs survey of the Magellanic System). The NIR photometry
was supplemented with well-sampled optical light curves and accurate pulsation
periods from the OGLE IV survey and the Gaia mission. We used the best-quality
NIR light curves to generate custom templates for modelling sparsely sampled
light curves in YJKs bands; in turn, we derived accurate and precise
intensity-averaged mean magnitudes and pulsation amplitudes of 339 T2Cs in the
MC. We used optical and NIR mean magnitudes to derive PL/PLC/PW relations for
T2Cs in multiple bands, which were calibrated with the geometric distance to
the LMC and with the Gaia parallaxes. We used our new empirical calibrations of
PL/PW relations to obtain distances to 22 T2C-host Galactic globular clusters,
which were found to be systematically smaller by 0.1 mag and 0.03-0.06 mag
compared with the literature. A better agreement is found between our distances
and those based on RRLs in globular clusters, providing strong support for
using these population II stars with the TRGB for future distance scale
studies.
