We present coordinated multiwavelength observations of the high Galactic latitude (b=+50 deg) black hole X-ray binary (XRB) J1357.2-0933 in quiescence. Our broadband spectrum includes strictly simultaneous radio and X-ray observations, and near-infrared, optical, and ultraviolet data taken 1-2 days later. We detect Swift J1357.2-0933 at all wavebands except for the radio (f_5GHz < 3.9 uJy/beam). Given current constraints on the distance (2.3-6.3 kpc), its 0.5-10 keV X-ray flux corresponds to an Eddington ratio Lx/Ledd = 4e-9 -- 3e-8 (assuming a black hole mass of 10 Msun). The broadband spectrum is dominated by synchrotron radiation from a relativistic population of outflowing thermal electrons, which we argue to be a common signature of short-period quiescent BHXBs. Furthermore, we identify the frequency where the synchrotron radiation transitions from optically thick-to-thin (approximately 2-5e14 Hz, which is the most robust determination of a jet break for a quiescent BHXB to date. Our interpretation relies on the presence of steep curvature in the ultraviolet spectrum, a frequency window made observable by the low amount of interstellar absorption along the line of sight. High Galactic latitude systems like Swift J1357.2-0933 with clean ultraviolet sightlines are crucial for understanding black hole accretion at low luminosities.
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