This paper deals with the existence of normalized solutions for a Schrödinger system of Choquard type with Sobolev critical nonlinearities \(\left\{ \begin{aligned} -\Delta u&= \lambda _1 u + \mu _1(I_\alpha *|u|^p)|u|^{p -2}u+\beta r_{1}|u|^{r_1-2}u|v|^{r_2}, \\ -\Delta v&= \lambda _2 v + \mu _2(I_\alpha *|v|^q)|v|^{q -2}v+\beta r_{2}|u|^{r_1}|v|^{r_2 -2}v \end{aligned} \right. \) and the restrictions \(\int _{\mathbb {R}^N}|u|^2\textrm{d}x=a\) and \(\int _{\mathbb {R}^N}|v|^2\textrm{d}x=b\) , where \(a,b>0\) are prescribed, \(N\in \{3,4\}\) , \(I_\alpha (x)\) is the Riesz potential, \(\alpha \in (0,N)\) , \(\mu _1\) , \(\mu _2\) , \(\beta >0\) , \(\frac{N+\alpha }{N}<p,q<\frac{N+\alpha +2}{N}\) , \(r_{1}\) , \(r_{2}>1\) and \(r_1+r_2=2^*:=\frac{2N}{N-2}\) . The frequencies \(\lambda _{1}\) and \(\lambda _{2}\) appear as Lagrange multipliers. We prove that the above system has a normalized ground state solution for \(0<\beta <\beta _0\) , where \(\beta _0\) is a constant.
