  Matching the number counts of high-$z$ sub-millimetre-selected galaxies
(SMGs) has been a long standing problem for galaxy formation models. In this
paper, we use 3D dust radiative transfer to model the sub-mm emission from
galaxies in the SIMBA cosmological hydrodynamic simulations, and compare
predictions to the latest single-dish observational constraints on the
abundance of 850$\mathrm{\mu m}$-selected sources. We find good agreement with
the shape of the integrated 850$\mathrm{\mu m}$ luminosity function, and the
normalisation is within 0.25 dex at $> 3 \; \mathrm{mJy}$, unprecedented for a
fully cosmological hydrodynamic simulation, along with good agreement in the
redshift distribution of bright SMGs. The agreement is driven primarily by
SIMBA's good match to infrared measures of the star formation rate (SFR)
function between $z = 2-4$ at high SFRs. Also important is the self-consistent
on-the-fly dust model in SIMBA, which predicts, on average, higher dust masses
(by up to a factor of 2.5) compared to using a fixed dust-to-metals ratio of
0.3. We construct a lightcone to investigate the effect of far-field blending,
and find that 52% of sources are blends of multiple components, which makes a
small contribution to the normalisation of the bright-end of the number counts.
We provide new fits to the 850$\mathrm{\mu m}$ luminosity as a function of SFR
and dust mass. Our results demonstrate that exotic solutions to the discrepancy
between sub-mm counts in simulations and observations, such as a top-heavy IMF,
are unnecessary, and that sub-millimetre-bright phases are a natural
consequence of massive galaxy evolution.
