The Antarctic Impulsive Transient Antenna (ANITA) experiment has observed two air shower events with energy $\sim 500~{\rm PeV}$ emerging from the Earth with exit angles $\sim 30^\circ$ above the horizon. As was immediately noted by the ANITA Collaboration, these events (in principle) could originate in the atmospheric decay of an upgoing $\tau$-lepton produced through a charged current interaction of a $\nu_\tau$ inside the Earth. However, the relatively steep arrival angles of these perplexing events create tension with the standard model (SM) neutrino-nucleon interaction cross section. Deepening the conundrum, the IceCube neutrino telescope and the Pierre Auger Observatory with substantially larger exposures to cosmic $\nu_\tau$'s in this energy range have not observed any events. This lack of observation implies that the messenger particle (MP) giving rise to ANITA events must produce an air shower event rate at least a factor of 40 larger than that produced by a flux of $\tau$-neutrinos to avoid conflicts with the upper limits reported by the IceCube and the Pierre Auger collaborations. In addition, the sensitivity of ANITA to MP-induced events must be comparable to or larger than those of IceCube and Auger to avoid conflict with the non-observation of any signal at these facilities. Beyond SM interpretations of ANITA events can be classified according to whether the MPs: (i) live inside the Earth, (ii) originate in neutrino-nucleon collisions inside the Earth, (iii) come from cosmological distances. In this communication we investigate the positive and negative facets of these three classes of models.