We present the discovery and characterization of a radio-bright binary in the Galactic globular cluster M10. First identified in deep radio continuum data from the Karl G. Jansky Very Large Array, M10-VLA1 has a flux density of 27 +/- 4 mu Jy at 7.4 GHz and a flat-to-inverted radio spectrum. Chandra imaging shows an X-ray source with L-X approximate to 10(31) erg s(-1) matching the location of the radio source. This places M10-VLA1 within the scatter of the radio-X-ray luminosity correlation for quiescent stellar-mass black holes, and a black hole X-ray binary is a viable explanation for this system. The radio and X-ray properties of the source disfavor, but do not rule out, identification as an accreting neutron star or white dwarf system. Optical imaging from the Hubble Space Telescope and spectroscopy from the SOAR telescope show that the system has an orbital period of 3.339 days and an unusual "red straggler" component: an evolved star found redward of the M10 red giant branch. These data also show UV/optical variability and double-peaked H alpha emission characteristic of an accretion disk. However, SOAR spectroscopic monitoring reveals that the velocity semi-amplitude of the red straggler is low. We conclude that M10-VLA1 is most likely either a quiescent black hole X-ray binary with a rather face-on (i < 4 degrees) orientation or an unusual flaring RS Canum Venaticorum variable-type active binary, and discuss future observations that could distinguish between these possibilities.