  Materials with very low thermal conductivity are of high interest for both
thermoelectric and optical phase-change applications. Synthetic nanostructuring
is most promising to suppress thermal conductivity by scattering phonons, but
challenges remain in producing bulk samples. We show that in crystalline
AgSbTe$_2$, 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 AgSbTe$_2$ 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 points to a new avenue in nano-scale
engineering of materials, to achieve low thermal conductivities for efficient
thermoelectric converters and phase-change memory devices.
