  We investigate the magnetic orders in a mixture of Boson (exciton) and
Fermion (electron or hole) trapped in transition-metal dichalcogenides moir\'e
superlattices. A sizable antiferromagnetic exchange interaction is found
between a carrier and an interlayer exciton trapped at different high symmetry
points of the moir\'e supercell. This interaction at a distance much shorter
than the carrier-carrier separation dominates the magnetic order in the
Bose-Fermi mixture, where the carrier sublattice develops ferromagnetism
opposite to that in the exciton sublattice. We demonstrate the possibility of
increasing the Curie temperature of moir\'e carriers through electrical tuning
of the exciton density in the ground state. In a trilayer moir\'e system with a
p-n-p type band alignment, the exciton-carrier interplay can establish a
layered antiferromagnetism for holes confined in the two outer layers. We
further reveal a spontaneous nematic order in the Bose-Fermi mixture, arising
from the interference between the Coulomb interaction and p-wave interlayer
tunneling dictated by the stacking registry.
