- yes, the Euclidean Jordan algebra approach to physics is fundamentally real in the sense that these algebras are algebras over ℝ and any element of such an algebra has a spectrum consisting of real numbers: it's a sum of real numbers times orthogonal minimal idempotents. So in this formalism one's access to the complex numbers is indirect. One way it shows up is that the automorphism group of the Euclidean Jordan algebra 𝕙ₙ(ℂ) has two connected components when n > 1, one consisting of maps
A ↦ UAU⁻¹
where A is self-adjoint and U is unitary, and one consisting of maps
A ↦ UAU⁻¹
where U is antiunitary. On the other hand the automorphism groups of the Jordan algebras 𝕙ₙ(ℝ), 𝕙ₙ(ℍ) and 𝕙₃(𝕆) are connected.
If one believes that what one has direct access to is measurements that are eigenvalues of self-adjoint operators, none of this is a bug.
As for the 'artificiality' of using only the connected component in the main theorem, the easiest way around it may be to not use only the connected component... and then show that the other component corresponds to CPT symmetry, and show that it's okay to gauge that - which would presumably only become noticeable if spacetime is not simply connected!
