An interesting feature of the finite-dimensional real spectral triple (A, H, D, J ) of the Standard Model is that it satisfies a "second-order" condition: conjugation by J maps the Clifford algebra C D (A) into its commutant, which in fact is isomorphic to the Clifford algebra itself (H is a self-Morita equivalence C D (A)-bimodule). This resembles a property of the canonical spectral triple of a closed oriented Riemannian manifold: there is a dense subspace of H which is a self-Morita equivalence C D (A)-bimodule. In this paper we argue that on manifolds, in order for the self-Morita equivalence to be implemented by a reality operator J , one has to introduce a "twist" and weaken one of the axioms of real spectral triples. We then investigate how the above mentioned conditions behave under products of spectral triples.

Twisted real structures are well-motivated as a way to implement the conformal transformation of a Dirac operator for a real spectral triple without needing to twist the noncommutative one-forms. We study the coupling of spectral triples with twisted real structures to gauge fields, adopting Morita equivalence via modules and bimodules as a guiding principle and paying special attention to modifications to the inner fluctuations of the Dirac operator. In particular, we analyze the twisted first-order condition as a possible alternative to abandoning the first-order condition in order to go beyond the standard model and elaborate upon the special case of gauge transformations accordingly. Applying the formalism to a toy model, we argue that under certain physically motivated assumptions, the spectral triple based on the left–right symmetric algebra should reduce to that of the standard model of fundamental particles and interactions, as in the untwisted case.