How to distinguish between the latently real matrices and the block quaternions?

Let a complex $n\times n$ matrix $A$ be unitarily similar to its entrywise conjugate matrix $\overline A$. If the unitary matrix $P$ in the relation $\overline A=P^*AP$ can be chosen symmetric (skew-symmetric), then $A$ is called a latently real matrix (respectively, a generalized block quaternion). Only these two cases are possible if $A$ is a (unitarily) irreducible matrix. The following question is discussed: How to find out whether the given $A$ is a latently real matrix or a generalized block quaternion?