Summary: In this paper, we consider the following chemotaxis-Stokes system with nonlinear diffusion and rotational flux \[ \begin{aligned} \left\{ \begin{array}{l} n_t+u\cdot \nabla n=\Delta n^m-\nabla \cdot (nS(x,n,c)\cdot \nabla c),\quad x\in \Omega , t>0,\\ c_t+u\cdot \nabla c=\Delta c-nc,\quad x\in \Omega , t>0,\\ u_t+\nabla P=\Delta u+n\nabla \phi ,\quad x\in \Omega , t>0,\\ \nabla \cdot u=0,\quad x\in \Omega , t>0\ \end{array} \right. \end{aligned} \tag{CNF} \] in a bounded domain \(\Omega \subseteq \mathbb{R}^3\) with smooth boundary, which describes the motion of oxygen-driven swimming bacteria in an incompressible fluid. Here the matrix-valued function \(S\in C^2(\bar{\Omega }\times [0,\infty )^2;\mathbb{R}^{3\times 3})\) fulfills \(|S(x,n,c)| \le S_0(c)\) for all \((x,n,c)\in \bar{\Omega } \times [0, \infty )\times [0, \infty )\) with \(S_0(c)\) nondecreasing on \([0,\infty )\). With developing some new methods (see Sect. 4 and Sect. 5), it is proved that under the condition \(m>\frac{10}{9}\) and proper regularity hypotheses on the initial data, the corresponding initial-boundary problem possesses at least one global weak solution, which is uniformly bounded. In view of \(S\) is a tensor-valued chemotactic sensitivity, it is easy to see that the restriction on \(m\) here is optimal (see Remark 3.1) and thus answer the open problem left in [N. Bellomo et al., Math. Models Methods Appl. Sci. 25, No. 9, 1663–1763 (2015; Zbl 1326.35397)] and [Y. Tao and M. Winkler, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 30, No. 1, 157–178 (2013; Zbl 1283.35154)]. This result significantly improves or extends previous results of several authors (see Remark 1.1).