We report on an upward traveling, radio-detected cosmic-ray-like impulsive event with characteristics closely matching an extensive air shower. This event, observed in the third flight of the Antarctic Impulsive Transient Antenna (ANITA), a NASA-sponsored long-duration balloon payload, is consistent with a similar event reported in a previous flight. These events may be produced by the atmospheric decay of an upward-propagating $tau$-lepton produced by a $ u_{tau}$ interaction, although their relatively steep arrival angles create tension with the standard model (SM) neutrino cross section. Each of the two events have $a~posteriori$ background estimates of $lesssim 10^{-2}$ events. If these are generated by $tau$-lepton decay, then either the charged-current $ u_{tau}$ cross section is suppressed at EeV energies, or the events arise at moments when the peak flux of a transient neutrino source was much larger than the typical expected cosmogenic background neutrinos.
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