  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.
