  In this paper, we investigate the following fractional Sobolev critical
nonlinear Schr\"{o}dinger (NLS) coupled systems: \begin{equation*}
\left\{\begin{array}{lll} (-\Delta)^{s} u=\mu_{1}
u+|u|^{2^{*}_{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^{*}_{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{array}\right. \end{equation*} where
$(-\Delta)^{s}$ is the fractional Laplacian, $N={3,4}$, $s\in(0,1)$, $\mu_{1},
\mu_{2}\in\mathbb{R}$ are unknown constants, which will appear as Lagrange
multipliers, $2^{*}_{s}$ is the fractional Sobolev critical index, $\eta_{1},
\eta_{2}, \gamma, m_{1}, m_{2}>0$, $\alpha>1, \beta>1$, $p, q,
\alpha+\beta\in(2+4s/N,2^{*}_{s}]$. Firstly, if $p, q, \alpha+\beta<2^{*}_{s}$,
we obtain the existence of positive normalized solution when $\gamma$ is big
enough. Secondly, if $p=q=\alpha+\beta=2^{*}_{s}$, 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.
