  (abridged)Variability is a key property of stars on the asymptotic giant
branch (AGB). Their pulsation period is related to the luminosity and mass-loss
rate of the star. The long-period variables (LPVs) and Mira variables are the
most prominent of all types of variability of evolved stars. The reddest, most
obscured AGB stars are too faint in the optical and have eluded large
variability surveys. Selection criteria are derived based on colour-colour and
colour-magnitude diagrams from the combination of VISTA Magellanic Cloud (VMC)
survey, Spitzer IRAC and AllWISE data. After eliminating LPVs with known
periods shorter than 450 days, a sample of 1299 candidate obscured AGB stars is
selected. K-band LCs are constructed combining the epoch photometry available
in the VMC survey with literature data, analysed for variability and fitted
with a single period sine curve to derive mean magnitudes, amplitudes and
periods. A subset of 254 stars are either new variables, or known variables
where the period we find is better determined than the literature value, or
longer than 1000 days. The spectral energy distributions (SEDs) of these stars
are fitted to a large number of templates. For this purpose the SEDs and
Spitzer IRS spectra of some non-AGB stars are also fitted to have templates of
the most likely contaminants in the sample. A sample of 217 likely LPVs is
found. Thirty-four stars have periods longer than 1000 days although some of
them have alternative shorter periods. Mass-loss rates and luminosities are
estimated from the template fitting. Period-luminosity relations are presented
for C- and O-rich Miras that appear to be extensions of relations derived in
the literature for shorter periods. The fit for the C-stars is particularly
well defined (with 182 objects) and reads Mbol = (-2.27 \pm 0.20) log P + (1.45
\pm 0.54)mag with an rms of 0.41 mag.
