  We aim to identify and characterise binary systems containing red supergiant
(RSG) stars in the Small Magellanic Cloud (SMC) using a newly available
ultraviolet (UV) point source catalogue obtained using the Ultraviolet Imaging
Telescope (UVIT) on board AstroSat. We select a sample of 560 SMC RSGs based on
photometric and spectroscopic observations at optical wavelengths and
cross-match this with the far-UV point source catalogue using the UVIT F172M
filter, finding 88 matches down to m$_{F172M}$=20.3 ABmag, which we interpret
as hot companions to the RSGs. Stellar parameters (luminosities, effective
temperatures and masses) for both components in all 88 binary systems are
determined and we find mass distributions in the ranges 6.1 to 22.3 Solar
masses for RSGs and 3.7 to 15.6 Solar masses for their companions. The most
massive RSG binary system in the SMC has a combined mass of 32 $\pm$4
M$_\odot$, with a mass ratio (q) of 0.92. By simulating observing biases, we
find an intrinsic multipliciy fraction of 18.8 $\pm$ 1.5% for mass ratios in
the range 0.3 < q < 1.0 and orbital periods approximately in the range 3 < log
P [days] < 8. By comparing our results with those of a similar mass on the
main-sequence, we determine the fraction of single stars to be ~20% and argue
that the orbital period distribution declines rapidly beyond log P ~ 3.5. We
study the mass-ratio distribution of RSG binary systems and find that a uniform
distribution best describes the data below 14 M$_\odot$. Above 15 M$_\odot$, we
find a lack of high mass-ratio systems.
