  We propose a model of asymmetric bosonic dark matter (DM) with self-repulsion
mediated by the vector field coupled to the complex scalar particles. By
adopting the two-fluid formalism, we study different DM distribution regimes,
either, fully condensed inside the core of a star or, otherwise, distributed in
a dilute halo around a neutron star (NS). We show that DM condensed in a core
leads to a decrease of the total gravitational mass, radius and tidal
deformability compared to a pure baryonic star with the same central density,
which we will perceive as an effective softening of the equation of state
(EoS). On the other hand, the presence of a DM halo increases the tidal
deformability and total gravitational mass. As a result, an accumulated DM
inside compact stars could mimic an apparent stiffening of strongly interacting
matter equation of state and constraints we impose on it at high densities.
  From the performed analysis of the effect of DM particles in a MeV-GeV
mass-scale, interaction strength, and relative DM fractions inside NSs we
obtained a rigorous constraint on model parameters. Finally, we discuss several
smoking guns of the presence of DM that are free from the above mentioned
apparent modification of the strongly interacting matter equation of state.
With this we could be probed with the future astrophysical and gravitational
wave (GW) surveys.
