The March 2018 report of two anomalous (Earth-emergent) e_cr ~ 0.6 EeVair showers by the ANITA collaboration has presented a severe puzzle of interpretation. Given existing limits on neutrino transient sources and the diffuse neutrino flux, these events are straightforwardly impossible under the Standard Model, due primarily to the challengesof propagating SM particles of these energies along the observed extended path lengths. I will review the nature of the ANITAexperiment and the properties of their anomalous events, and presentour arguments for the incompatibility of these observations with the Standard Model. I will also explore the possible existence of analog events among the highest-energy neutrinos of the IceCube neutrinoobservatory. Together, these observations encourage us to consider the chief alternative, that the ANITA Anomalous Events are mediated by arelatively long-lived Beyond Standard Model (BSM) particle. I will present our inferences as to the necessary properties of thisparticle, which appear consistent (at least in part) with those predicted for the "stau" slepton in some supersymmetric modelsof the fundamental interactions. LHC Run 2 data now under analysis may prove to be highly relevant in this context.
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