  We propose an expansion of the definition of almost-commutative spectral
triple that accommodates non-trivial fibrations and is stable under inner
fluctuation of the metric, and then prove a reconstruction theorem for
almost-commutative spectral triples under this definition as a simple
consequence of Connes's reconstruction theorem for commutative spectral
triples. Along the way, we weaken the orientability hypothesis in the
reconstruction theorem for commutative spectral triples, and following
Chakraborty and Mathai, prove a number of results concerning the stability of
properties of spectral triples under suitable perturbation of the Dirac
operator.
