Multiparticle Quantum Mechanics with independent phases

Question

I am pondering a variant of quantum mechanics, where each particle actually has, loosely speaking, 'its own' complex phase. A two-particle wave-function would thus be written as e.g.

$\psi(x,y)=e^{i_x x p_1 }e^{i_y y p_2 }$

where $i_x$ and $i_y$ are independent. $\psi$ thus takes values not in $\mathbb C$, but rather in  $\mathbb C \otimes_{\mathbb R}\mathbb C$. The tensor product has to be taken over $\mathbb R$, otherwise the 'effect' obviously vanishes.

A two particle state now contains more (local) phase information than in the standard. I have not done any strict calculations, but surprisingly it does not seem to make much difference for actual systems like the Helium atom or for scattering experiments.

What differences would there be to standard QM? (I have not been able to find any paper of a similar approach or calculation)