  Let X be a locally compact Polish space and let m be a reference Radon
measure on X. Let $\Gamma_X$ denote the configuration space over X, that is,
the space of all locally finite subsets of X. A point process on X is a
probability measure on $\Gamma_X$. A point process $\mu$ is called
determinantal if its correlation functions have the form
$k^{(n)}(x_1,\ldots,x_n)=\det[K(x_i,x_j)]_{i,j=1,\ldots,n}$. The function
K(x,y) is called the correlation kernel of the determinantal point process
$\mu$. Assume that the space X is split into two parts: $X=X_1\sqcup X_2$. A
kernel K(x,y) is called J-Hermitian if it is Hermitian on $X_1\times X_1$ and
$X_2\times X_2$, and $K(x,y)=-\overline{K(y,x)}$ for $x\in X_1$ and $y\in X_2$.
We derive a necessary and sufficient condition of existence of a determinantal
point process with a J-Hermitian correlation kernel K(x,y).
