  We investigate the holonomy group of singular K\"ahler-Einstein metrics on
klt varieties with numerically trivial canonical divisor. Finiteness of the
number of connected components, a Bochner principle for holomorphic tensors,
and a connection between irreducibility of holonomy representations and
stability of the tangent sheaf are established. As a consequence, known
decompositions for tangent sheaves of varieties with trivial canonical divisor
are refined. In particular, we show that up to finite quasi-\'etale covers,
varieties with strongly stable tangent sheaf are either Calabi-Yau or
irreducible holomorphic symplectic. These results form one building block for
H\"oring-Peternell's recent proof of a singular version of the
Beauville-Bogomolov Decomposition Theorem.
