  We present a homogeneously-selected sample of 15779 candidate binary systems
with main sequence primary stars and orbital periods shorter than 5 days. The
targets were selected from TESS full-frame image lightcurves on the basis of
their tidally-induced ellipsoidal modulation. Spectroscopic follow-up suggests
a sample purity of $83 \pm 13$ per cent. Injection-recovery tests allow us to
estimate our overall completeness as $28 \pm 3$ per cent with $P_\mathrm{orb} <
3$ days and to quantify our selection effects. We estimate that $39 \pm 4$ per
cent of our sample are contact binary systems, and we disentangle the period
distributions of the contact and detached binaries. We derive the orbital
period distribution of the main sequence binary population at short orbital
periods, finding a distribution continuous with the log-normal distribution
previously found for solar-type stars at longer periods, but with a significant
steepening at $P_\mathrm{orb} \lesssim 3$ days, and a pile-up of contact
binaries at $P_\mathrm{orb} \approx 0.4$ days. Companions in the period range
1--5 days are found to be an order of magnitude more frequent around stars
hotter than $\approx 6250 K$ (the Kraft break) when compared to cooler stars,
suggesting that magnetic braking plays an important role in shaping the
temperature and period distributions. We detect resolved tertiary companions to
$9.0 \pm 0.2$ per cent of our binaries with a median separation of 3200 AU. The
frequency of tertiary companions rises to $29 \pm 5$ per cent among the systems
with the shortest ellipsoidal periods. This large binary sample with quantified
selection effects will be a powerful resource for future studies of detached
and contact binary systems with $P_\mathrm{orb} < 5$ days.
