  We show that a general semi-annihilation scenario, in which a pair of dark
matter (DM) particles annihilate to an anti-DM, and an unstable state that can
mix with or decay to standard model states, can lead to particle anti-particle
asymmetry in the DM sector. The present DM abundance, including the
CP-violation in the DM sector and the resulting present asymmetry are
determined entirely by a single semi-annihilation process at next-to-leading
order. For large CP-violation in this process, we find that a nearly complete
asymmetry can be obtained in the DM sector, with the observed DM density being
dominated by the (anti-)DM particle. The presence of additional
pair-annihilation processes can modify the ratio of DM and anti-DM number
densities further, if the pair-annihilation is active subsequent to the
decoupling of the semi-annihilation. For such a scenario, the required
CP-violation for generating the same present asymmetry is generically much
smaller, as compared to the scenario with only semi-annihilation present. We
show that a minimal model with a complex scalar DM with cubic self-interactions
can give rise to both semi- and pair-annihilations, with the required
CP-violation generated at one-loop level. We also find that the upper bound on
the DM mass from S-matrix unitarity in the purely asymmetric semi-annihilation
scenario, with maximal CP-violation, is around 15 GeV, which is much stronger
than in the WIMP and previously considered asymmetric DM cases, due to the
required large non-zero chemical potential for such asymmetric DM.
