Rephrased and tried to clarify further.

@DieterKadelka clarified.

Thanks, this is very helpful. Do you have a reference for this approach using invariant tensors for each group? I'm still having a little trouble following how to formulate the problem this way.

It is not clear that it must always be impossible though. Perhaps there is some N and some non-trivial (and non-Gaussian) random matrices that satisfy $\phi(A_1A_2A_1A_2) - \phi(A_1)^2\phi(A_2^2) - \phi(A_1^2)\phi(A_2)^2+\phi(A_1)^2\phi(A_2)^2=0$

I think OP is asking if this is true for any fixed $N$.

Chapter 2 of "Schmüdgen, Konrad. An invitation to unbounded representations of*-algebras on Hilbert space. Cham: Springer, 2020." covers a lot of material on *-algebras over R, properties that do/do-not transfer through their complexification, etc.