When does $det begin{pmatrix} A & X \ X^T & A end{pmatrix} = (det A)^2 + (det X)^2$?

Let $A$ be an $n times n$ real symmetric matrix.
Let
$$
M = begin{pmatrix} A & X \ X^T & A end{pmatrix}
$$
where $X$ is a real invertible $n times n$ matrix. I am interested in finding $X$ such that
$$
det M = (det A)^2 + (det X)^2.
$$

Given $A$, is there a systematic way of finding such $X$ under some hypotheses?

Any suggestions, comments or reference for a related topic is very much appreciated. Thank you!