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In hybrid materials, a high-symmetry lattice is decorated by low-symmetry building blocks. The result is an aperiodic solid that hosts many nearly-degenerate disordered configurations. Using the perovskite methylammonium lead iodide (MAPbI 3 ) as a prototype hybrid material, we show that the inherent disorder renders the conventional phonon picture of transport insufficient. Ab initio molecular dynamics and analysis of the spectral energy density reveal that vibrational carriers simultaneously exhibit features of both classical phonons and of carriers typically found in glasses. The low frequency modes retain elements of acoustic waves but exhibit extremely short lifetimes of only a few tens of picoseconds. For higher frequency modes, strong scattering due to rapid motion and reconfiguration of the organic cation molecules induces a loss of definition of the wave vector. Lattice dynamics shows that these carriers are more akin to diffusons – the nonwave carriers in vitreous materials – and are the dominant contributors to thermal conduction near room temperature. To unify the framework of glassy diffusons with that of phonons scattered at the ultimate limit, three-phonon interactions resolved from first-principles expose anharmonic effects two orders of magnitude higher than in silicon. The dominant anharmonic interactions occur within modes of the PbI 6 octahedral framework itself, as well as between modes of the octahedral framework and modes localized to the MA molecules. The former arises from long-range interactions due to resonant bonding, and the latter from polar rotor scattering of the MA molecules. This establishes a clear microscopic connection between symmetry-breaking, dynamical disorder, anharmonicity, and the loss of wave nature in MAPbI 3 .