  We consider noncommutative GUT inspired field theories formulated within the
enveloping-algebra formalism for anomaly safe compact simple gauge groups. Our
theories have only gauge fields and fermions, and we compute the UV divergent
part of the one-loop background-field effective action involving two fermionic
fields at first order in the noncommutativity parameter theta. We show that, if
the second-degree Casimir has the same value for all the irreps furnished by
the fermionic multiplets of the model, then, that UV divergent part can be
renormalised by carrying out multiplicative renormalisations of the coupling
constant, theta and the fields, along with the inclusion of theta-dependent
counterterms which vanish upon imposing the equations of motion. These
theta-dependent counterterms have no physical effect since they vanish
on-shell. This result along with the vanishing of the UV divergent part of the
fermionic four-point functions leads to the unexpected conclusion that the
one-loop matter sector of the background-field effective action of these
theories is one-loop multiplicatively renormalisable on-shell. We also show
that the background-field effective action of the gauge sector of the theories
considered here receives no theta-dependent UV divergent contributions at
one-loop. We thus conclude that these theories are on-shell one-loop
multiplicatively renormalisable at first order in theta.
