Neutron stars in low-mass binaries can accrete matter and angular momentum from a non-degenerate companion star and be spun-up to rapid spin periods, making them detectable as millisecond radio pulsars (MSPs). Although the swollen stellar companion transfers mass onto the neutron star for hundreds of millions of years or more, most MSPs in the Galactic field have a compact white dwarf companion in a wide orbit, representing the end product of the spin-up process. Following the launch of the \emph{Fermi} Space Telescope, it became clear MSPs were nearly ubiquitous emitters of high-energy $\gamma$-rays. Using multiwavelength follow-up observations of unidentified \emph{Fermi} sources, the rarely observed binary MSPs with non-degenerate companions started being discovered in greater numbers. In this dissertation, I expand this population of rare MSPs with new discoveries and present a novel method for identifying and characterizing new Galactic compact neutron star binaries: cross-matching unassociated Fermi sources with short-period optical variables from high-cadence all-sky surveys. This technique has allowed for the discovery of a number of new systems with non-degenerate companions in orbits around previously unknown MSPs. Among these are a few systems with red giant companions in relatively wide orbits ($P\gtrsim1$ d) that will naturally evolve into the MSP--white dwarfs that represent the bulk of known MSP binaries. The long orbital periods, giant companions, and inferred evolutionary tracks of these systems indicate they are entirely different from the close-orbit MSP binaries being found recently with \emph{Fermi}, and suggests a new subclass of compact neutron star systems that are the progenitors of typical field MSP binaries.
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