  We prove that the factorization of a saturated fusion system over a discrete
$p$-toral group as a product of indecomposable subsystems is unique up to
normal automorphisms of the fusion system and permutations of the factors. In
particular, if the fusion system has trivial center, or if its focal subgroup
is the entire Sylow group, then this factorization is unique (up to the
ordering of the factors). This result was motivated by questions about
automorphism groups of products of fusion systems.
