As I commented elsewhere, in the multi particle state example you can fix the problem by tensoring with respect to the vector or matrix algebra, so it works jus

As I commented elsewhere, in the multi particle state example you can fix the problem by tensoring with respect to the vector or matrix algebra, so it works just fine. And like of course it does because C itself is isomorphic to these algebras. My point about calling into question what one means by “using complex numbers” is that the article says it is “necessary to establish some ground rules that exclude real-valued quantum theories that restate the standard complex-valued theory by other names” but then handwaves the rules. Presumably you’re only allowed to tensor over R. But more importantly, it provides no motivation for why these ground rules wouldn’t be somewhat arbitrary. I guess I just don’t see the value of removing complex numbers from the formulation.