  We present the complete next-to-leading order calculation of isolated prompt
photon production in association with a jet in deep-inelastic scattering. The
calculation involves, direct, resolved and fragmentation contributions. It is
shown that defining the transverse momenta in the proton virtual-photon frame
(CM*), as usually done, or in the laboratory frame (LAB), as done in some
experiments, is not equivalent and leads to important differences concerning
the perturbative approach. In fact, using the latter frame may preclude, under
certain conditions, the calculation of the next-to-leading order correction to
the important resolved component. A comparaison with the latest ZEUS data is
performed and good agreement is found in the perturbatively stable regions.
