We report the discovery of a new low-mass X-ray binary near the center of the unassociated Fermi GeV gamma-ray source 4FGL J0540.0-7552. The source shows the persistent presence of an optical accretion disk and exhibits extreme X-ray and optical variability. It also has an X-ray spectrum well-fit by a hard power law with a Gamma = 1.8 and a high ratio of X-ray to gamma-ray flux. Together, these properties are consistent with the classification of the binary as a transitional millisecond pulsar (tMSP) in the sub-luminous disk state. Uniquely among the candidate tMSPs, 4FGL J0540.0-7552 shows consistent optical, X-ray, and gamma-ray evidence for having undergone a state change, becoming substantially brighter in the optical and X-rays and fainter in GeV gamma-rays sometime in mid-2013. In its current sub-luminous disk state, and like one other candidate tMSP in the Galactic field, 4FGL J0540.0-7552 appears to always be in an X-ray flare mode, indicating that this could be common phenomenology for tMSPs.
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