  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 $\nu_{\tau}$ 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 $\nu_{\tau}$ 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.
