where ω is the phonon frequency, v is the group velocity, τ -1 is the scattering rate, and n is the phonon occupation for phonon modes λ and λ ′ . The diagonal part, λ = λ ′ , denotes the Peierls term describing particle-like transport that reduces to the usual Boltzmann transport formulation, while the off-diagonal part, λ ̸ = λ ′ , denotes the Wigner term describing the coherent tunneling between distinct phonon modes. The Wigner contribution increases relative to the Peierl contribution as modes become closer in energy and their broadening due to scattering increases such that they significantly overlap. As such, the Wigner term manifests substantially in materials with very soft phonons and heavy anharmonic scattering, e.g. CsPbBr 3 , Cu 12 Sb 3 S 13 , and increasingly so at higher temperatures. Eq. 6 is evaluated using FourPhonon-ShengBTE with in-house modification to incorporate the Wigner contribution (see Code Availability for details). Anharmonic scattering is limited to three-phonon processes arising from third-order IFCs as this is known to yield accurate κ lat for most materials.