  We have discovered a new candidate redback millisecond pulsar binary near the
center of the error ellipse of the bright unassociated Fermi-LAT $\gamma$-ray
source 4FGL J0940.3-7610. The candidate counterpart is a variable optical
source that also shows faint X-ray emission. Optical photometric and
spectroscopic monitoring with the SOAR telescope indicates the companion is a
low-mass star in a 6.5-hr orbit around an invisible primary, showing both
ellipsoidal variations and irradiation and consistent with the properties of
known redback millisecond pulsar binaries. Given the orbital parameters,
preliminary modeling of the optical light curves suggests an edge-on
inclination and a low-mass ($\sim 1.2$ - $1.4\,M_{\odot}$) neutron star, along
with a secondary mass somewhat more massive than typical $\gtrsim
0.4\,M_{\odot}$. This combination of inclination and secondary properties could
make radio eclipses more likely for this system, explaining its previous
non-discovery in radio pulsation searches. Hence 4FGL J0940.3-7610 may be a
strong candidate for a focused search for $\gamma$-ray pulsations to enable the
future detection of a millisecond pulsar.
