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A $ Z_3 $ symmetric 3-Higgs Doublet Model (3HDM) with two inert doublets andone active doublet (that plays the role of the Higgs doublet), the so-calledI(2+1)HDM, is studied. We discuss the behaviour of this 3HDM realisation whenone allows for a $ Z_3 $ soft breaking term. Such a symmetry enables thepresence of a two-component Dark Matter (DM) scenario in the form of"Hermaphrodite DM", where the two inert candidates have opposite CP parity andare protected by this discrete symmetry from decaying into Standard Model (SM)particles. Furthermore, the two DM states are potentially distinguishable fromeach other as they cannot be subsumed into a complex field, having differentmasses and gauge couplings. With this in mind, we study differential spectrawith a distinctive shape from which the existence of two different DM componentdistributions could be easily inferred. We prove this to be possible at theLarge Hadron Collider (LHC) via the $q\bar{q} \to 2l+H_1H_1$ and $q\bar{q} \to2l+A_1A_1$ processes as well as at a future electron-positron machine via the$e^{+}e^{-} \to 2l+H_1H_1$ and $e^{+}e^{-} \to 2l+A_1A_1$ channels, where$l=e,\mu$.

The $Z_3$ soft breaking in the I(2+1)HDM and its probes at present and future colliders