ABSTRACT We exploit EAGLE, a cosmological hydrodynamical simulation, to reproduce the selection of the observed submillimetre (submm) galaxy population by selecting the model galaxies at z ≥ 1 with mock submm fluxes $S_{850\, μ \mathrm{m}}$ ≥ 1 mJy. We find a reasonable agreement between the model galaxies within this sample and the properties of the observed submm population, such as their star formation rates (SFRs) at z < 3, redshift distribution, and many integrated galaxy properties. We find that the median redshift of the $S_{850\, μ \mathrm{m}}$ ≥ 1 mJy model population is z ≈ 2.5, and that they are massive galaxies (M* ∼ 1011 M⊙) with high dust masses (Mdust ∼ 108 M⊙), gas fractions (fgas ≈ 50 per cent), and SFRs ($\dot{M}_* ≈ 100$ M⊙ yr−1). In addition, we find that they have major and minor merger fractions similar to the general population, suggesting that mergers are not the sole driver of the high SFRs in the model submm galaxies. Instead, the $S_{850\, μ \mathrm{m}}$ ≥ 1 mJy model galaxies yield high SFRs primarily because they maintain a significant gas reservoir as a result of hosting an undermassive black hole relative to comparably massive galaxies. Not all ‘highly star-forming’ ($\dot{M}_* \ge 80$ M⊙ yr−1) eagle galaxies have submm fluxes $S_{850\, μ \mathrm{m}}$ ≥ 1 mJy. We investigate the nature of these highly star-forming ‘Submm-Faint’ galaxies (i.e. $\dot{M}_* \ge 80$ M⊙ yr−1 but $S_{850\, μ \mathrm{m}}$ < 1 mJy) and find that they are similar to the model submm galaxies, being gas rich and hosting undermassive black holes. However, they are also typically at higher redshifts (z > 4) and are lower mass (M* ∼ 1010 M⊙). These typically higher redshift galaxies show stronger evidence for having been triggered by major mergers, and critically, they are likely missed by most current submm surveys due to their higher dust temperatures and lower dust masses.