  We extend the results about left-invariant Codazzi tensor fields on Lie
groups equipped with left-invariant Riemannian metrics obtained by d'Atri in
1985 to the setting of reductive homogeneous spaces $G/H$, where the curvature
of the canonical connection of second kind associated with the fixed reductive
decomposition $\mathfrak{g} = \mathfrak{h}\oplus\mathfrak{m}$ enters the
picture. In particular, we show that invariant Codazzi tensor fields on a
naturally reductive homogeneous space are parallel.
