  The LIGO-Virgo-KAGRA Collaboration recently detected gravitational waves
(GWs) from the merger of black-hole-neutron-star (BHNS) binary systems GW200105
and GW200115. No coincident electromagnetic (EM) counterparts were detected.
While the mass ratio and BH spin in both systems were not sufficient to tidally
disrupt the NS outside of the BH event horizon, other, magnetospheric
mechanisms for EM emission exist in this regime and depend sensitively on the
NS magnetic field strength. Combining GW measurements with EM flux upper
limits, we place upper limits on the NS surface magnetic field strength above
which magnetospheric emission models would have generated an observable EM
counterpart. We consider fireball models powered by the black-hole battery
mechanism, where energy is output in gamma-rays over $\lesssim1$~second.
Consistency with no detection by Fermi-GBM or INTEGRAL SPI-ACS constrains the
NS surface magnetic field to $\lesssim10^{15}$~G. Hence, joint GW detection and
EM upper limits rule out the theoretical possibility that the NSs in GW200105
and GW200115, and the putative NS in GW190814, retain $\gtrsim10^{15}$~G
dipolar magnetic fields until merger. They also rule out formation scenarios
where strongly magnetized magnetars quickly merge with BHs. We alternatively
rule out operation of the BH-battery powered fireball mechanism in these
systems. This is the first multi-messenger constraint on NS magnetic fields in
BHNS systems and a novel approach to probe fields at this point in NS
evolution. This demonstrates the constraining power that multi-messenger
analyses of BHNS mergers have on BHNS formation scenarios, the magnetic-field
evolution in NSs, and the physics of BHNS magnetospheric interactions.
