Let $B \in \mathbb{C}^{n \times n}$ be a Hermitian matrix, $B^{R}$ and $B^{I}$ its real and imaginary parts, respectively. It can be converted to a real-symmetric matrix via this construction. $$ \left[\begin{array}{cc} B^{R} & -B^{I} \\ B^{I} & B^{R} \end{array}\right] $$
Now given a general real-symmetric matrix of $2n\times2n$ dimensions, it cannot always be converted to a Hermitian matrix. It needs to satisfy the block structure shown above.
Is it possible to impose compose additional constraints on the real-symmetric matrix so that it can be converted to Hermitian?
Examples
A Hermitian matrix,
$$\left( \begin{array}{cc} 0.497633\, +0. i & 0.0967618\, -0.490542 i \\ 0.0967618\, +0.490542 i & 0.502367\, +0. i \\ \end{array} \right)$$
converted to a real-symmetric matrix, $$\left( \begin{array}{cccc} 0.497633 & 0.0967618 & 0. & 0.490542 \\ 0.0967618 & 0.502367 & -0.490542 & 0. \\ 0. & -0.490542 & 0.497633 & 0.0967618 \\ 0.490542 & 0. & 0.0967618 & 0.502367 \\ \end{array} \right)$$
A real-symmetric matrix, $$\left( \begin{array}{cccc} 0.325682 & -0.0419613 & 0.177914 & 0.715459 \\ -0.0419613 & 0.00540634 & -0.0229227 & -0.0921806 \\ 0.177914 & -0.0229227 & 0.0971914 & 0.390842 \\ 0.715459 & -0.0921806 & 0.390842 & 1.57172 \\ \end{array} \right)$$ that cannot be converted to Hermitian.
Let $V$ be a finite dimensional complex inner product space, with orthonormal basis $v_1, \ldots, v_n$. Your conversion of complex Hermitian matrices to real matrices corresponds to the basis of $V_{\mathbb R}$, which is the name I'll give to $V$ considered as a real vector space, given by $$v_1, \ldots, v_n, i v_1, \ldots, i v_n.$$
$V_\mathbb R$ has a canonical inner product induced by the same norm that $V$ has (maybe there's a better way to define the inner product here, but the Polarization identity does the trick), along with a $90^\circ$ rotation map $R: V_\mathbb R \to V_\mathbb R$ given by multiplication by $i$. The Hermitian matrices over $V$ correspond precisely to the symmetric matrices over $V_\mathbb R$ which commute with the operator $R$.
The definition of $R$ under your choice of basis is given by $$\begin{pmatrix} 0 & -I_n \\ I_n & 0\end{pmatrix}.$$
What this ends up translating to is simply, if the real $2n \times 2n$ matrix is given in block form by $$\begin{pmatrix} A & B \\ C & D\end{pmatrix},$$ then to come from a complex Hermitian matrix requires $A=A^T=D=D^T$ and $B = C^T=-C$.