Summary: In this paper we investigate the existence of the positive normalized solutions for the coupled Hartree-Fock type nonlocal elliptic system \[ \begin{cases} -\Delta u_1 -\lambda_1 u_1 & = \mu_1\left( \int_{\mathbb{R}^N}\frac{|u_1 (y)|^{p_1}}{|x-y|^{N-\alpha}}dy\right) |u_1|^{p_1 -2}u_1 \\ & \quad +\frac{\beta r_1}{r_1 +r_2} \left( \int_{\mathbb{R}^N}\frac{| u_2 (y)|^{r_2}}{|x-y|^{N-\alpha}}dy\right) |u_1|^{r_1 -2}u_1, \\ -\Delta u_2 -\lambda_2 u_2 & = \mu_2\left( \int_{\mathbb{R}^N}\frac{| u_2 (y)|^{p_2}}{|x-y|^{N-\alpha}}dy\right) |u_2|^{p_2 -2}u_2 \\ & \quad +\frac{\beta r_2}{r_1 +r_2} \left( \int_{\mathbb{R}^N}\frac{| u_1 (y)|^{r_1}}{|x-y|^{N-\alpha}}dy\right) |u_2|^{r_2 -2}u_2, \\ \int_{\mathbb{R}^N}|u_1|^2 dx & =a_1^2, \quad \int_{\mathbb{R}^N}|u_2|^2 dx=a_2^2, N\geq 3, \end{cases} \] where \(a_1,a_2, \mu_1,\mu_2,\beta >0\), and the constants \(\lambda_1\) and \(\lambda_2\) are unknown and will appear as Lagrange multipliers. First, we prove the existence of two positive solutions when \(N=3\), \(p_1 =p_2 =\alpha =2\) and \(\frac{N+\alpha +2}{N}<r_1, r_2 <\frac{N+\alpha}{N-2}\). Precisely, the first solution is a local minimizer for which we establish the compactness of the minimizing sequences when \(N\geq 3\), \(\alpha \in (0,N)\), \(\frac{N+\alpha}{N}<p_1\), \(p_2<\frac{N+\alpha +2}{N}\) and \(\frac{N+\alpha +2}{N}<r_1\), \(r_2<\frac{N+\alpha}{N-2}\). The second one is obtained by mountain pass argument in the special case \(N=3\), \(p_1 =p_2 =\alpha =2\) and \(\frac{N+\alpha +2}{N}<r_1\), \(r_2<\frac{N+\alpha}{N-2}\). Second, we study the existence of two positive solutions by the compactness of the minimizing sequences and linking type procedure when \(N=5\), \(p_1 =p_2 =\alpha =2\) and \(\frac{N+\alpha}{N}<r_1\), \(r_2<\frac{N+\alpha +2}{N}\). In both the cases, we assume that \(\beta >0\) is sufficiently small. Finally, by using the mountain pass argument, we consider the existence of a positive solution of the system when \(\alpha \in (\max \{0,N-4\},N)\), \(\frac{N+\alpha}{N}<p_1 <\frac{N+\alpha +2}{N}<p_2\), \(r_1 <\frac{N+\alpha}{N-2}\) and \(\frac{N+\alpha +4}{N}<r_2 <\frac{N+\alpha}{N-2}\).