  We investigate the microscopic mechanism of ultralow lattice thermal
conductivity ($\kappa_l$) of TlInTe$_2$ and its weak temperature dependence
using a unified theory of lattice heat transport that considers contributions
arising from the particle-like propagation as well as wave-like tunneling of
phonons. While we use the Peierls-Boltzmann transport equation (PBTE) to
calculate the particle-like contributions ($\kappa_l$(PBTE)), we explicitly
calculate the off-diagonal (OD) components of the heat-flux operator within a
first-principles density functional theory framework to determine the
contributions ($\kappa_l$(OD)) arising from the wave-like tunneling of phonons.
At each temperature, T, we anharmonically renormalize the phonon frequencies
using the self-consistent phonon theory including quartic anharmonicity, and
utilize them to calculate $\kappa_l$(PBTE) and $\kappa_l$(OD). With the
combined inclusion of $\kappa_l$(PBTE), $\kappa_l$(OD), and additional
grain-boundary scatterings, our calculations successfully reproduce the
experimental results. Our analysis shows that large quartic anharmonicity of
TlInTe$_2$ (a) strongly hardens the low-energy phonon branches, (b) diminishes
the three-phonon scattering processes at finite T, and (c) recovers the weaker
than T$^{-1}$ decay of the measured $\kappa_l$.
