  We propose a new physics scenario in which the decay of a very heavy
dark-matter candidate which does not interact with the neutrino sector could
explain the two anomalous events recently reported by ANITA. The model is
composed of two components of dark matter, an unstable dark-sector state, and a
massive dark gauge boson. We assume that the heavier dark-matter particle of
EeV-range mass is distributed over the galactic halo and disintegrates into a
pair of lighter -- highly boosted -- dark-matter states in the present universe
which reach and penetrate the Earth. The latter scatters {\it in}elastically
off a nucleon and produces a heavier dark-sector unstable state which
subsequently decays back to the lighter dark matter along with hadrons, which
induce Extensive Air Showers, via on-/off-shell dark gauge boson. Depending on
the mass hierarchy within the dark sector, either the dark gauge boson or the
unstable dark-sector particle can be long-lived, hence transmitted
significantly through the Earth. We study the angular distribution of the
signal and show that our model favors emergence angles in the range $\sim
25^\circ -35^\circ$ if the associated parameter choices bear the situation
where the mean free path of the boosted incident particle is much larger than
the Earth diameter while its long-lived decay product has a decay length of
dimensions comparable to the Earth radius. Our model, in particular, avoids any
constraints from complementary neutrino searches such as IceCube or the Auger
observatory.
