Pseudo-commutation classes of complex matrices and their decomplexification

The relation between complex matrices $H$ and $A$ given by the equality $HA=\overline{A}H$ is called the pseudo-commutation. The set $S_H$ of all $A$ that pseudo-commute with a nonsingular $n\times n$ matrix $H$ is called the pseudo-commutation class defined by $H$. Every class $S_H$ is a subspace of the space $M_n(\mathbf{C})$ interpreted as a real vector space of dimension $2n^2$. Under the assumption $\mathrm{dim}_{\mathbf{R}}S_H=n^2$, we find a necessary and sufficient condition for the possibility to decomplexify all the matrices in $S_H$ by one and the same similarity transformation.