Materials with very low thermal conductivity are of great interest for both thermoelectric and optical phase-change applications. Synthetic nanostructuring is most promising for suppressing thermal conductivity through phonon scattering, but challenges remain in producing bulk samples. In crystalline AgSbTe₂ we show that a spontaneously forming nanostructure leads to a suppression of thermal conductivity to a glass-like level. Our mapping of the phonon mean free paths provides a novel bottom-up microscopic account of thermal conductivity and also reveals intrinsic anisotropies associated with the nanostructure. Ground-state degeneracy in AgSbTe₂ leads to the natural formation of nanoscale domains with different orderings on the cation sublattice, and correlated atomic displacements, which efficiently scatter phonons. This mechanism is general and suggests a new avenue for the nanoscale engineering of materials to achieve low thermal conductivities for efficient thermoelectric converters and phase-change memory devices.