Then what is "quantum theory based on the real numbers"? I think your notion of "real amplitudes" cannot be the complete answer because the Schrodinger equation

Then what is "quantum theory based on the real numbers"? I think your notion of "real amplitudes" cannot be the complete answer because the Schrodinger equation is linear, and complex linear algebra is just a special case of real linear algebra. I looked at the appendix of the paper and their claim seems to hinge on some form of non-decomposibility of a certain tensor product state? It is the paragraph below equation A2. More generally I got a bit frustrated reading the paper because the axioms of real quantum mechanics did not seem to be properly formulated.