  A mixed graph $G$ is a graph obtained from a simple undirected graph by
orientating a subset of edges. $G$ is self-converse if it is isomorphic to the
graph obtained from $G$ by reversing each directed edge. For two mixed graphs
$G$ and $H$ with Hermitian adjacency matrices $A(G)$ and $A(H)$, we say $G$ is
$\mathbb{R}$\emph{-cospectral} to $H$ if, for any $y\in \mathbb{R}$, $yJ-A(G)$
and $yJ-A(H)$ have the same spectrum, where $J$ is the all-one matrix. A
self-converse mixed graph $G$ is said to be determined by its generalized
spectrum, if any self-converse mixed graph that is $R$-cospectral with $G$ is
isomorphic to $G$. Let $G$ be a self-converse mixed graph of order $n$ such
that $2^{-\lfloor n/2\rfloor}\det W$ (which is always a real or pure imaginary
Gaussian integer) is square-free in $\mathbb{Z}[i]$, where
$W=[e,Ae,\ldots,A^{n-1}e]$, $A=A(G)$ and $e$ is the all-one vector. We prove
that, for any self-converse mixed graph $H$ that is $\mathbb{R}$-cospectral to
$G$, there exists a Gaussian rational unitary matrix $U$ such that $Ue=e$,
$U^*A(G)U=A(H)$ and $(1+i)U$ is a Gaussian integral matrix. In particular, if
$G$ is an ordinary graph (viewed as a mixed graph) satisfying the above
condition, then any self-converse mixed graph $H$ that is
$\mathbb{R}$-cospectral to $G$ is $G$ itself (in the sense of isomorphism).
This strengthens a recent result of the first author.
