  We explore factorizations of noncommutative Riemannian spin geometries over
commutative base manifolds in unbounded KK-theory. After setting up the general
formalism of unbounded KK-theory and improving upon the construction of
internal products, we arrive at a natural bundle-theoretic formulation of gauge
theories arising from spectral triples. We find that the unitary group of a
given noncommutative spectral triple arises as the group of endomorphisms of a
certain Hilbert bundle; the inner fluctuations split in terms of connections
on, and endomorphisms of, this Hilbert bundle. Moreover, we introduce an
extended gauge group of unitary endomorphisms and a corresponding notion of
gauge fields. We work out several examples in full detail, to wit Yang--Mills
theory, the noncommutative torus and the $\theta$-deformed Hopf fibration over
the two-sphere.
