  Halide perovskite (HP) semiconductors exhibit unique strong coupling between
the electronic and structural dynamics. The high-temperature cubic phase of HPs
is known to be entropically stabilized, with imaginary frequencies in the
calculated phonon dispersion relation. Similar calculations, based on the
static average crystal structure, predict a stable tetragonal phase with no
imaginary modes. This work shows that in contrast to standard theory
predictions, the room-temperature tetragonal phase of CH$_{3} $NH$_{3}
$PbI$_{3}$ is strongly anharmonic. We use Raman polarization-orientation (PO)
measurements and \textit{ab initio} molecular dynamics (AIMD) to investigate
the origin and temperature evolution of the strong structural anharmonicity
throughout the tetragonal phase. Raman PO measurements reveal a new spectral
feature that resembles a soft mode. This mode shows an unusual continuous
increase in damping with temperature which is indicative of an anharmonic
potential surface. The analysis of AIMD trajectories identifies two major
sources of anharmonicity: the orientational unlocking of the [CH$_{3}
$NH$_{3}$]$^+$ ions and large amplitude octahedral tilting that continuously
increases with temperature. Our work suggests that the standard phonon picture
cannot describe the structural dynamics of tetragonal CH$_{3} $NH$_{3}
$PbI$_{3}$.
