  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.
