  After the successful detection of a gravitational-wave (GW) signal and its
associated electromagnetic (EM) counterparts from GW170817, neutron star--black
hole (NSBH) mergers have been highly expected to be the next type of
multi-messenger source. However, despite the detection of several of NSBH
merger candidates during the GW third observation run, no confirmed EM
counterparts from these sources have been identified. The most plausible
explanation is that these NSBH merger candidates were plunging events mainly
because the primary BHs had near-zero projected aligned-spins based on GW
observations. In view that NSs can be easily tidally disrupted by BHs with high
projected aligned-spins, we study an evolution channel to form NSBH binaries
with fast-spinning BHs, the properties of BH mass and spin, and their
associated tidal disruption probability. We find that if the NSs are born
firstly, the companion helium stars would be tidally spun up efficiently, and
would thus finally form fast-spinning BHs. If BHs do not receive significant
natal kicks at birth, these NSBH binaries that can merge within the Hubble time
would have BHs with the projected aligned-spins $\chi_{z}\gtrsim0.8$ and,
hence, can certainly allow tidal disruption to happen. Even if significant BH
kicks are considered for a small fraction of NSBH binaries, the projected
aligned-spins of BHs are $\chi_z\gtrsim0.2$. These systems can still be
disrupted events unless the NSs are very massive. Thus, NS-first-born NSBH
mergers would be promising multi-messenger sources. We discuss various
potential EM counterparts associated with these systems and their detectability
in the upcoming fourth observation run.
