The aim of this paper is to establish multiple positive normalized solutions \((u,v,\lambda _1,\lambda _2)\in H^1(\mathbb {R}^N,\mathbb {R}^2)\times \mathbb {R}^2\) to the following coupled Schrödinger system involving Sobolev critical exponent: \( {\left\{ \begin{array}{ll} -\Delta u+\lambda _1 u=\mu _1|u|^{p-2}u+\nu \alpha |u|^{\alpha -2}u|v|^\beta , x\in \mathbb {R}^N,\\ -\Delta v+\lambda _2 v=\mu _2|v|^{q-2}v+\nu \beta |v|^{\beta -2}v|u|^\alpha , x\in \mathbb {R}^N,\\ \int _{\mathbb {R}^N}|u|^2\textrm{d}x=a, \int _{\mathbb {R}^N}|v|^2\textrm{d}x=b, \end{array}\right. } N\ge 3, \) where \(\mu _1,\mu _2, \nu , a, b>0\) . We are particularly interested in the mass mixed case that \(2<p, q<2+\frac{4}{N}, \alpha>1, \beta >1\) , and \(\alpha +\beta =2^*:=\frac{2N}{N-2}\) . For sufficiently small \(\nu >0\) , we demonstrate that the above system admits two positive solutions, one of which serves as a local minimizer, and the other as a mountain pass solution. By developing some new technical lemmas on the interaction estimates, we are managed to resolves Soave’s open problem [J. Funct. Anal., 2020, Remark 1.1] within the context of the system case. Notably, our existence result holds true for all dimensions \(N\ge 3\) . Our results also significantly extend the result of Gou and Jeanjean [Nonlinearity, 2018, Theorem 1.1] to the Sobolev critical coupled case and removing the hypothesis “either \(p,q\le \alpha +\beta -\frac{2}{N}\) or \(|p-q|\le \frac{2}{N}\) " for \(N\ge 5\) . Additionally, we also establish a sequence of properties for the local minimizer, including local uniqueness, continuity with respect to the small parameter \(\nu \) , and the limiting profiles for \(\nu \rightarrow 0^+\) .
