  We classify the 5-dimensional homogeneous geometries in the sense of
Thurston. The present paper (part 3 of 3) classifies those in which the linear
isotropy representation is nontrivial but reducible. Most of the resulting
geometries are products. Some interesting examples include a countably infinite
family of inequivalent geometries diffeomorphic to $S^3 \times S^2$; an
uncountable family in which only a countable subfamily admits compact
quotients; and the non-maximal geometry SO(4)/SO(2) realized by two distinct
maximal geometries.
