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We compare the contribution of active galactic nuclei (AGNs) and star formation towards dust heating in sub-mm galaxies (SMGs). We have used ALMA at 0.1-arcsec resolution to image a complete flux-limited sample of seven sub-mm sources previously shown to have spectral energy distributions that were as well-fitted by obscured AGN as star-forming galaxy templates. Indeed, two sub-mm sources were known to be quasars from their absorbed X-ray emission. We find the sub-mm sizes of all SMGs to be small (≈1−2 kpc) and generally ∼3 times smaller than any host detected in the near-infrared (NIR). In all cases, the five SMGs are comparable in sub-mm size to the two known quasars and four z ≈ 6 quasars, also observed with ALMA. We detect no evidence of diffuse spiral arms in this complete sample. We then convert the far-infrared (FIR) luminosities to star formation rate (SFR) surface densities and find that the SMGs occupy the same range as the known quasars in our sample. We conclude that in terms of sub-mm size, extent relative to host and SFR density as well as luminosity and mid-IR (MIR) colour, there is little distinction between the SMGs and sub-mm bright quasars. Finally, we present preliminary evidence that SMGs with higher MIR luminosities and sub-mm loud quasars tend to have dust components that range to hotter temperatures than their less luminous SMG counterparts. In light of these results, we continue to suggest that luminous SMGs may host dust-absorbed quasars that may simultaneously dominate the FIR and hard X-ray backgrounds.