We present coordinated multiwavelength observations of the high Galactic latitude (b=+50 degrees) black holeX-ray binary (BHXB) Swift J1357.2-0933 in quiescence. Our broad-band spectrum includes strictly simultaneous radio and X-ray observations, and near-infrared, optical, and ultraviolet data taken 1-2 d later. We detect Swift J1357.2-0933 at all wavebands except for the radio (f(5 GHz) < 3.9 mu Jy beam(-1); 3 sigma(rms)). Given current constraints on the distance (2.3-6.3 kpc), its 0.5-10 keV X-ray flux corresponds to an Eddington ratio L-X/L-Edd = 4 x 10(-9)-3 x 10(-8) (assuming a black hole mass of 10 M-circle dot). The broad-band 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 (nu(b) approximate to 2-5 x 10(14) 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.