Summary: In this paper, we investigate the following fractional Sobolev critical Nonlinear Schrödinger coupled systems: \[ \begin{cases} (-\Delta)^s u = \mu_1 u+|u|^{2^\ast_s-2}u + \eta_1|u|^{p-2}u + \gamma\alpha |u|^{\alpha -2}u|v|^{\beta } &\text{in }\mathbb{R}^N,\\ (-\Delta)^s v = \mu_2 v+|v|^{2^\ast_s-2}v + \eta_2|v|^{q-2}v + \gamma\beta |u|^\alpha|v|^{\beta -2}v &\text{in }\mathbb{R}^N,\\ \|u\|^2_{L^2} = m_1^2 \text{ and } \|v\|^2_{L^2} = m_2^2, \end{cases} \] where \((-\Delta)^s\) is the fractional Laplacian, \(N > 2s\), \(s\in (0, 1)\), \(\mu_1, \mu_2\in\mathbb{R}\) are unknown constants, which will appear as Lagrange multipliers, \(2^\ast_s\) is the fractional Sobolev critical index, \(\eta_1, \eta_2, \gamma, m_1, m_2 > 0\), \(\alpha>1, \beta > 1\), \(p, q, \alpha + \beta\in(2+4s/N, 2^\ast_s]\). Firstly, if \(p, q, \alpha +\beta <2^\ast_s\), we obtain the existence of positive normalized solution when \(\gamma\) is big enough. Secondly, if \(p=q=\alpha +\beta=2^\ast_s\), we show that nonexistence of positive normalized solution. The main ideas and methods of this paper are scaling transformation, classification discussion and concentration-compactness principle.