  The ANITA collaboration have reported observation of two anomalous events
that appear to be $\varepsilon_{\rm cr} \approx 0.6$ EeV cosmic ray showers
emerging from the Earth with exit angles of $27^\circ$ and $35^\circ$,
respectively. While EeV-scale upgoing showers have been anticipated as a result
of astrophysical tau neutrinos converting to tau leptons during Earth passage,
the observed exit angles are much steeper than expected in Standard Model (SM)
scenarios. Indeed, under conservative extrapolations of the SM interactions,
there is no particle that can propagate through the Earth with probability $p >
10^{-6}$ at these energies and exit angles. We explore here whether "beyond the
Standard Model" (BSM) particles are required to explain the ANITA events, if
correctly interpreted, and conclude that they are. Seeking confirmation or
refutation of the physical phenomenon of sub-EeV Earth-emergent cosmic rays in
data from other facilities, we find support for the reality of the ANITA
events, and three candidate analog events, among the Extremely High Energy
Northern Track neutrinos of the IceCube Neutrino Observatory. Properties of the
implied BSM particle are anticipated, at least in part, by those predicted for
the "stau" slepton ($\tilde{\tau}_R$) in some supersymmetric models of the
fundamental interactions, wherein the stau manifests as the next-to-lowest mass
supersymmetric partner particle.
