  We refine the reconstruction theorem for almost-commutative spectral triples
to a result for real almost-commutative spectral triples, clarifying, in the
process, both concrete and abstract definitions of real commutative and
almost-commutative spectral triples. In particular, we find that a real
almost-commutative spectral triple algebraically encodes the commutative
*-algebra of the base manifold in a canonical way, and that a compact oriented
Riemannian manifold admits real (almost-)commutative spectral triples of
arbitrary KO-dimension. Moreover, we define a notion of smooth family of real
finite spectral triples and of the twisting of a concrete real commutative
spectral triple by such a family, with interesting KK-theoretic and
gauge-theoretic implications.
