  We use deep ground-based imaging in the near-IR to search for counterparts to
the luminous submm sources in the catalog of Smail et al (1998). For the
majority of the submm sources the near-IR imaging supports the counterparts
originally selected from deep optical images. However, in two cases (10% of the
sample) we find a relatively bright near-IR source close to the submm position,
sources that were unidentified in the deep HST and ground-based R-band images
used in Smail et al (1998). We place limits on colours of these sources from
deep high-resolution Keck II imaging and find they have 2-sigma limits of
(I-K)>6.8 and (I-K)>6.0 respectively. Both sources thus class as EROs. Using
the spectral properties of the submm source in the radio and submm we argue
that these EROs are probably the source of the submm emission, rather than the
bright spiral galaxies previously identified by Smail et al. (1998). From the
surface density of these submm-bright EROs we suggest that this class accounts
for the majority of the reddest members of the ERO population, in good
agreement with the preliminary conclusions of pointed submm observations of
individual EROs. We conclude that the most extreme EROs represent a population
of dusty, ultraluminous galaxies at high redshifts; further study of these will
provide insights into the nature of star formation in obscured galaxies in the
early Universe. The identification of similar counterparts in blank field submm
surveys will be extremely difficult owing to their faintness (K~20.5, I>26.5).
Finally, we discuss the radio and submm properties of the two submm-bright EROs
discovered here and suggest that both galaxies lie at z>2.
