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.
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