  ALICE is one of four large experiments at the CERN Large Hadron Collider near
Geneva, specially designed to study particle production in ultra-relativistic
heavy-ion collisions. Located 52 meters underground with 28 meters of
overburden rock, it has also been used to detect muons produced by cosmic ray
interactions in the upper atmosphere. In this paper, we present the
multiplicity distribution of these atmospheric muons and its comparison with
Monte Carlo simulations. This analysis exploits the large size and excellent
tracking capability of the ALICE Time Projection Chamber. A special emphasis is
given to the study of high multiplicity events containing more than 100
reconstructed muons and corresponding to a muon areal density $\rho_{\mu} >
5.9~$m$^{-2}$. Similar events have been studied in previous underground
experiments such as ALEPH and DELPHI at LEP. While these experiments were able
to reproduce the measured muon multiplicity distribution with Monte Carlo
simulations at low and intermediate multiplicities, their simulations failed to
describe the frequency of the highest multiplicity events. In this work we show
that the high multiplicity events observed in ALICE stem from primary cosmic
rays with energies above $10^{16}$ eV and that the frequency of these events
can be successfully described by assuming a heavy mass composition of primary
cosmic rays in this energy range. The development of the resulting air showers
was simulated using the latest version of QGSJET to model hadronic
interactions. This observation places significant constraints on alternative,
more exotic, production mechanisms for these events.
