  We study the coupled Hartree system $$ \left\{\begin{array}{ll} -\Delta u+
V_1(x)u =\alpha_1\big(|x|^{-4}\ast u^{2}\big)u+\beta \big(|x|^{-4}\ast
v^{2}\big)u &\mbox{in}\ \mathbb{R}^N,\\[1mm] -\Delta v+ V_2(x)v
=\alpha_2\big(|x|^{-4}\ast v^{2}\big)v +\beta\big(|x|^{-4}\ast u^{2}\big)v
&\mbox{in}\ \mathbb{R}^N, \end{array}\right. $$ where $N\geq 5$,
$\beta>\max\{\alpha_1,\alpha_2\}\geq\min\{\alpha_1,\alpha_2\}>0$, and
$V_1,\,V_2\in L^{N/2}(\mathbb{R}^N)\cap L_{\text{loc}}^{\infty}(\mathbb{R}^N)$
are nonnegative potentials. This system is critical in the sense of the
Hardy-Littlewood-Sobolev inequality. For the system with $V_1=V_2=0$ we employ
moving sphere arguments in integral form to classify positive solutions and to
prove the uniqueness of positive solutions up to translation and dilation,
which is of independent interest. Then using the uniqueness property, we
establish a nonlocal version of the global compactness lemma and prove the
existence of a high energy positive solution for the system assuming that
$|V_1|_{L^{N/2}(\mathbb{R}^N)}+|V_2|_{L^{N/2}(\mathbb{R}^N)}>0$ is suitably
small.
