Modelling dust formation in single stars evolving through the carbon-star stage of the asymptotic giant branch (AGB) reproduces well the mid-infrared colours and magnitudes of most of the C-rich sources in the Large Magellanic Cloud (LMC), apart from a small subset of extremely red objects (EROs). An analysis of the spectral energy distributions of EROs suggests the presence of large quantities of dust, which demand gas densities in the outflow significantly higher than expected from theoretical modelling. We propose that binary interaction mechanisms that involve common envelope (CE) evolution could be a possible explanation for these peculiar stars; the CE phase is favoured by the rapid growth of the stellar radius occurring after C/O overcomes unity. Our modelling of the dust provides results consistent with the observations for mass-loss rates $\dot{M} \sim 5\times 10^{-4}\,{\rm M}_{\odot }$ yr-1, a lower limit to the rapid loss of the envelope experienced in the CE phase. We propose that EROs could possibly hide binaries with orbital periods of about days and are likely to be responsible for a large fraction of the dust production rate in galaxies.