We present a study of PSR J1723-2837, an eclipsing, 1.86 ms millisecond binary radio pulsar discovered in the Parkes Multibeam survey. Radio timing indicates that the pulsar has a circular orbit with a 15 hr orbital period, a low-mass companion, and a measurable orbital period derivative. The eclipse fraction of ~15% during the pulsars orbit is twice the Roche lobe size inferred for the companion. The timing behavior is significantly affected by unmodeled systematics of astrophysical origin, and higher-order orbital period derivatives are needed in the timing solution to account for these variations. We have identified the pulsars (non-degenerate) companion using archival ultraviolet, optical, and infrared survey data and new optical photometry. Doppler shifts from optical spectroscopy confirm the stars association with the pulsar and indicate a pulsar-to-companion mass ratio of 3.3 +/- 0.5, corresponding to a companion mass range of 0.4 to 0.7 Msun and an orbital inclination angle range of between 30 and 41 degrees, assuming a pulsar mass range of 1.4-2.0 Msun. Spectroscopy indicates a spectral type of G for the companion and an inferred Roche-lobe-filling distance that is consistent with the distance estimated from radio dispersion. The features of PSR J1723-2837 indicate that it is likely a redback system. Unlike the five other Galactic redbacks discovered to date, PSR J1723-2837 has not been detected as a gamma-ray source with Fermi. This may be due to an intrinsic spin-down luminosity that is much smaller than the measured value if the unmeasured contribution from proper motion is large.
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