I have a misunderstanding that I am hoping is really quite trivial.
In connes standard Non-commutative geometry model of electroweak interactions he takes the algebra input in his finite spectral triple to be $A_F=\mathbb{C}⊕\mathbb{H}$. He then tensors this finite algebra with the algebra of complex functions over a smooth manifold $C^\infty(M)$.
As far as I am aware this tensor product must be over the complex numbers, but the quaternions $\mathbb{H}$ are a real algebra. It is impossible to centralize the complex numbers as a sub algebra of the quaternions. For this reason I am wondering how his tensor product over $\mathbb{C}$ is well defined.
Question: How is tensoring two spectral triples over the complex numbers well defined when a real algebra is chosen?