What is the Schmidt basis equivalent for a mixed state?

A pure state can be locally transformed unitarily to its Schmidt basis given by

$$|psirangle_{AB} = sum_i alpha_i |irangle_A otimes |irangle_B$$

where $A$ and $B$ are both diagonalized, and can be seen to have the same spectrum.

For a mixed state $rho_{AB}$, A and B do not have the same spectrum, but its subsystems $rho_{A(B)} = mathrm{Tr}_{B(A)} rho_{AB}$ should both be diagonalizable at the same time, what is such a basis?