Suppose I have a symmetric matrix A, and several diagonal matrices $D_1,D_2,...$

Are there any matrix transformations, such as $P^\top A P$ so that $P^\top AP$, $P^\top D_1 P$, $P^\top D_2 P$, etc are either all tridiagonal, or all have minimimal bandwidth in some sense? If for example, I only had $D_1$ then solving the generalized eigenvalue problem for the matrix pencil $A,D_1$ would give me a basis that simultaneously diagonalizes both $A$ and $D_1$. Obviously, one basis will not simultaneously diagonalize more matrices in general, but a set of banded matrices would also be pretty nice.

I am aware of Tisseur and Garvey's papers on simultaneous tridiagonalization of two matrices.