According to Gardiner-Zoller (Quantum Noise), operators acting on the density matrix can be mapped via e.g. (I'm taking Wigner space as an example, but the same holds for P and Q)
$$a\rho\leftrightarrow\left(\alpha+\frac{1}{2}\frac{\partial}{\partial\alpha^*}\right)W( \alpha,\alpha^*)$$ $$\rho a^\dagger\leftrightarrow\left(\alpha^*+\frac{1}{2}\frac{\partial}{\partial \alpha}\right) W(\alpha,\alpha^*)$$
Below, an example is given using the P-function, frow which it is clear that if multiple operators are applied on the left or the right of the density matrix, the same correspondences hold, as long as the operators closest to $\rho$ are applied first (i.e. the phase-space representation most to the right, so closest to $W$).
Now my question is: what if there are operators acting on both sides of $\rho$? In the simplest case of $a\rho a^\dagger$ this does not seem to be an issue, because $\left(\alpha+\frac{1}{2}\frac{\partial}{\partial\alpha^*}\right)\left(\alpha^*+\frac{1}{2}\frac{\partial}{\partial \alpha}\right)$ =$\left(\alpha^*+\frac{1}{2}\frac{\partial}{\partial \alpha}\right)\left(\alpha+\frac{1}{2}\frac{\partial}{\partial\alpha^*}\right)$,
but it does lead to ambiguity for example for $aa\rho a^\dagger a^\dagger$.
I would expect that the proper way of doing this is still from the inside out, alternating operators from the left and the right; also because this way I obtain a result that is real. Is this correct? How to properly see this?