  [abridged] The properties of galaxies with the reddest observed R-K colors
(Extremely Red Objects, EROs), including their apparent division into passive
and obscured active objects with roughly similar number densities, are a known
challenge for models of galaxy formation. We produce mock catalogues generated
by interfacing the predictions of the semi-analytical MORGANA model for the
evolution of galaxies in a Lambda-CDM cosmology with the spectro-photometric +
radiative transfer code GRASIL and Infrared (IR) template library to show that
the model correctly reproduces number counts, redshift distributions and active
fractions of R-K>5 sources. We test the robustness of our results against
different dust attenuations and, most importantly, against the inclusion of
TP-AGB stars in Simple Stellar Populations used to generate galaxy spectra, and
find that the inclusion of TP-AGBs has a relevant effect, in that it allows to
increase by a large factor the number of very red active objects at all color
cuts. We find that though the most passive and the most obscured active
galaxies have a higher probability of being selected as EROs, many EROs have
intermediate properties and the population does not show bimodality in specific
star formation rate (SSFR). We predict that deep observations in the Far-IR,
from 100 to 500 micron, are the most efficient way to constrain the SSFR of
these objects; we give predictions for future Herschel observations. Finally,
we test whether a simple evolutionary sequence for the formation of z=0 massive
galaxies, going through a sub-mm-bright phase and then a ERO phase, are typical
in this galaxy formation model. We find that this sequence holds for ~25 per
cent of z=0 massive galaxies, while the model typically shows a more complex
connection between sub-mm, ERO and massive galaxies. [abridged]
