Let $T_1$ and $S_1$ be bounded, self-adjoint operators on a Hilbert space $H_1$, and let $T_2$ and $S_2$ be bounded, self-adjoint operators on a Hilbert space $H_2$ such that the tensor products
$$T := T_1 \otimes T_2 \quad \text{and} \quad S := S_1 \otimes S_2$$
commute (while $T_1$ and $S_1$, as well as $T_2$ and $S_2$, may not commute). I was wondering if some version of the spectral theorem (in its multiplication operator form) applies in this case respecting the tensor product structure. That is, I would like to know whether it holds that there exist measure spaces $(X_1, \Sigma_1, \mu_1)$ and $(X_2, \Sigma_2, \mu_2)$, unitaries $U_1 : L^2(\mu_1) \to H_1$ and $U_2 : L^2(\mu_2) \to H_2$ and functions $f,g \in L^\infty (\mu_1 \otimes \mu_2)$ such that
$$(U_1 \otimes U_2)^{-1} \, T \, (U_1 \otimes U_2) = M_f \, , \\ (U_1 \otimes U_2)^{-1} \, S \, (U_1 \otimes U_2) = M_g \, , $$
where $M_f, M_g$ are multiplication operators on $L^2(\mu_1 \otimes \mu_2)$.