We present coordinated multiwavelength observations of the high Galactic latitude (b = +50°) black hole X-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 (f5 GHz < 3.9 μJy beam-1; 3σrms). 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 = 4 × 10-9-3 × 10-8 (assuming a black hole mass of 10 M⊙). 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 (νb ≈ 2-5 × 1014 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.