When is $\left[\begin{smallmatrix} D_1 & B \\\\ -B^T & D_2 \end{smallmatrix} \right]$ $\mathbb{R}$-diagonalizable?

Question

Is there some block-wise characterization of $\mathbb{R}$-diagonalizability (by similarities) of

$$\begin{bmatrix} D_1 & B \\\\ -B^T & D_2 \end{bmatrix},$$

where $D_1$ and $D_2$ are real diagonal, and $B$ is real (generally rectangular)?