  The promise by the LIGO/Virgo/Kagra (LVK) collaboration to detect black
hole--neutron star (BH--NS) mergers via gravitational wave (GW) emission has
recently been fulfilled with the detection of GW200105 and GW200115. Mergers of
BH--NS binaries are particularly exciting for their multi-messenger potential,
since the GW detection can be followed by an electromagnetic (EM) counterpart
(kilonova, gamma-ray burst, afterglow) that can reveal important information on
the equation of state (EOS) of NSs and the nature of the BH spin. We carry out
a statistical study of the binary stars that evolve to form a BH--NS binary and
compute the rate of merger events that can be followed by an EM counterpart. We
find that $\gtrsim 50\%$ of the mergers can lead to an EM counterpart only in
the case BHs are born highly spinning ($\chi_{\rm BH}\gtrsim 0.7$), while this
fraction does not exceed about $30\%$ for stiff NS EOSs and a few percent for
soft NS EOSs for low-spinning BHs ($\chi_{\rm BH}\lesssim 0.2$), suggesting
that a high rate of EM counterparts of BH--NS would provide support for high
natal BH spins. However, the possibilities that BHs are born with near-maximal
spins and that NS internal structure is described by a stiff EOS are disfavored
by current LVK constraints. Considering that these values only represent an
upper limit to observe an EM counterpart due to current observational
limitations, as in brightness sensitivity and sky localization, BH--NS mergers
are unlikely multi-messenger sources.
