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We report new simultaneous X-ray and radio continuum observations of 3FGL J0427.9-6704, a candidate member of the enigmatic class of transitional millisecond pulsars. These XMM-Newton and Australia Telescope Compact Array observations of this nearly edge-on, eclipsing low-mass X-ray binary were taken in the sub-luminous disk state at an X-ray luminosity of ~10^33(d/2.3kpc)^2 erg/s. Unlike the few well-studied transitional millisecond pulsars, which spend most of their disk state in a characteristic high or low accretion mode with occasional flares, 3FGL J0427.9-6704 stayed in the flare mode for the entire X-ray observation of ~20 hours, with the brightest flares reaching ~2*10^34 erg/s. The source continuously exhibited flaring activity on time-scales of ~10-100 sec in both the X-ray and optical/UV. No measurable time delay between the X-ray and optical/UV flares is observed, but the optical/UV flares last longer, and the relative amplitudes of the X-ray and optical/UV flares show a large scatter. The X-ray spectrum can be well-fit with a partially-absorbed power-law (Gamma~1.4-1.5), perhaps due to the edge-on viewing angle. Modestly variable radio continuum emission is present at all epochs, and is not eclipsed by the secondary, consistent with the presence of a steady radio outflow or jet. The simultaneous radio/X-ray luminosity ratio of 3FGL J0427.9-6704 is higher than any known transitional millisecond pulsars and comparable to that of stellar-mass black holes of the same X-ray luminosity, providing additional evidence that some neutron stars can be as radio-loud as black holes.