  Let $X$ be a compact K\"ahler space with klt singularities and vanishing
first Chern class. We prove the Bochner principle for holomorphic tensors on
the smooth locus of $X$: any such tensor is parallel with respect to the
singular Ricci-flat metrics. As a consequence, after a finite quasi-\'etale
cover $X$ splits off a complex torus of the maximum possible dimension. We then
proceed to decompose the tangent sheaf of $X$ according to its holonomy
representation. In particular, we classify those $X$ which have strongly stable
tangent sheaf: up to quasi-\'etale covers, these are either irreducible
Calabi--Yau or irreducible holomorphic symplectic. As an application of these
results, we show that if $X$ has dimension four, then it satisfies Campana's
Abelianity Conjecture.
