  It was recently pointed out that direct detection signals from at least three
different targets may be used to determine whether the Dark Matter (DM)
particle is different from its antiparticle. In this work, we examine in detail
the feasibility of this test under different conditions, motivated by proposals
for future detectors. Specifically, we perform likelihood fits to mock data
under the hypotheses that the DM particle is identical to or different from its
antiparticle, and determine the significance with which the former can be
rejected in favor of the latter. In our analysis, we consider 3 different
values of the DM mass ($50$ GeV, $300$ GeV, $1$ TeV) and 4 different
experimental ensembles, each consisting of at least 3 different targets -- Xe
and Ar plus one among the following: Si, Ge, $\mathrm{CaWO_4}$, or
Ge/$\mathrm{CaWO_4}$. For each of these experimental ensembles and each DM
mass, the expected discrimination significance is calculated as a function of
the DM-nucleon couplings. In the best case scenario, the discrimination
significance can reach $\mathcal{O}(3\sigma)$ for three of the four ensembles
considered, and $\mathcal{O}(5\sigma)$ for the ensemble including
$\mathrm{Si}$, highlighting the need for a variety of experimental targets in
order to determine the DM properties. These results show that future direct
detection signals could be used to exclude, at a statistically significant
level, a Majorana or a real DM particle, giving a critical clue about the
identity of the Dark Matter.
