  We consider compact conformal manifolds $(M,[g])$ endowed with a closed Weyl
structure $\nabla$, i.e. a torsion-free connection preserving the conformal
structure, which is locally but not globally the Levi-Civita connection of a
metric in $[g]$. Our aim is to classify all such structures when both $\nabla$
and $\nabla^g$, the Levi-Civita connection of $g$, have special holonomy. In
such a setting, $(M,[g],\nabla)$ is either flat, or irreducible, or carries a
locally conformally product (LCP) structure. Since the flat case is already
completely classified, we focus on the last two cases. When $\nabla$ has
irreducible holonomy we prove that $(M,g)$ is either Vaisman, or a mapping
torus of an isometry of a compact nearly K\"ahler or nearly parallel
$\mathrm{G}_2$ manifold, while in the LCP case we prove that $g$ is neither
K\"ahler nor Einstein, thus reducible by the Berger-Simons Theorem, and we
obtain the local classification of such structures in terms of adapted metrics.
