  We discuss the compatibility of the combined annual modulation effect
measured by DAMA/LIBRA-phase1 and DAMA/LIBRA-phase2 with an explanation in
terms of inelastic scattering events induced by the most general
Galilean-invariant effective contact interaction of a Weakly Interacting
Massive Particle (WIMP) dark matter particle of spin 0, 1/2 or 1. We take into
account all the possible interferences among operators by studying the
intersections among the ellipsoidal surfaces of constant signal of DAMA and
other experiments in the space of the coupling constants of the effective
theory. In our analysis we assume a standard Maxwellian velocity distribution
in the Galaxy. We find that, compared to the elastic case, inelastic scattering
partially relieves but does not eliminate the existing tension between the DAMA
effect and the constraints from the null results of other experiments. Such
tension is very large in all the parameter space with the exception of a small
region for WIMP mass $m_{\chi}\simeq$ 10 GeV and mass splitting $\delta\gtrsim$
20 keV, where it is partially, but not completely relieved. In such region the
bounds from fluorine targets are evaded in a kinematic way because the minimal
WIMP incoming speed required to trigger upscatters off fluorine exceeds the
maximal WIMP velocity in the Galaxy, or is very close to it. As a consequence,
we also find that the residual tension between DAMA and other results is more
sensitive on the astrophysical parameters compared to the elastic case. We find
that the configurations with the smallest tension can produce enough yearly
modulation in some of the DAMA bins in compliance with the constraints from
other experiments, but the ensuing shape of the modulation spectrum is too
steep compared to the measured one. For such configurations the recent
COSINE-100 bound is evaded in a natural way due to their large expected
modulation fractions.
