A $2n\times 2n$ dimensional Hermitian matrix that can be diagonalized by a symplectic transformation can be viewed as an $n\times n$ matrix with elements consisting of $2\times 2$ blocks of the quaternion real form
${\bar{z}\;-\bar{w}}\choose{w\;\; z}$
so if you choose real $z$ and $w$ you have constructed a real symmetric matrix $M$ that can be diagonalized by a symplectic $S$; is S$.
@Federico: this what you need?is the general form for matrices that commute, $MT=TM$, with
$T=1_{N}\otimes$ ${0\; 1}\choose{-1\; 0}$ $K$
($K$ is the operator of complex conjugation); alternatively, one can take matrices that anticommute, $MT=-TM$; then the $2\times 2$ blocks have the form
${\bar{z}\;\bar{w}}\choose{w\;\; -z}$
and again, for a real $M$ one would choose real $w,z$. these two choices exhaust the possibilities.
In applications to physical systems, the matrix $M$ is a Hamiltonian and $T$ is the operator of time reversal. Then only commuting matrices, $MT=TM$, are permitted.
For a discussion in the physics context, see Section 1.4.2 of Forrester's book, online here:
