  We report on the X-ray, optical, near-infrared, submillimetre and radio
properties of five Extremely Red Objects (EROs) selected at X-ray wavelengths
by XMM-Newton in the Lockman Hole field. They all have enough counts in the
X-ray band to allow spectral fitting: four are most probably obscured,
Compton-thin AGN with redshift dependent absorbing column densities of 10^22 -
10^24 cm-2, whilst the fifth is best fitted by a thermal spectrum and is likely
to be a massive elliptical galaxy in a deep gravitational potential. Their
optical/near-infrared colours and sizes suggest that X-ray selected EROs
comprise a mixture of dusty 'starburst' galaxies and non-dusty galaxies that
are dominated by either star-light or light from an active nucleus. The colour
diagnostics are supported by the submillimetre and radio data: the two AGN with
'starburst' colours have submillimetre or radio flux densities that imply large
star-formation rates, whilst those with 'elliptical' colours do not. The one
source detected in the submillimetre waveband has narrow emission lines at a
redshift of 1.45. Although the bulk of its radio emission originates from
processes other than star formation, it is most probably a radio-quiet
ultraluminous infrared galaxy.
