Summary: This paper deals with the following chemotaxis-Stokes system \[ \begin{cases} n_t+u\cdot \nabla n=\Delta n^m-\nabla \cdot (nS(x,n,c)\cdot \nabla c), \quad & x\in \Omega ,\,\, t>0,\\ c_t+u\cdot \nabla c=\Delta c-nf(c),\quad & x\in \Omega , \,\,t>0,\\ u_t=\Delta u+\nabla P+n\nabla \phi ,\quad & x\in \Omega , \,\,t>0,\\ \nabla \cdot u=0, \quad & x\in \Omega , \,\,t>0 \end{cases} \] under no-flux boundary conditions in a bounded domain \(\Omega \subset \mathbb {R}^{3}\) with smooth boundary, where \(m\geq 1\), \(\phi \in W^{1,\infty }(\Omega )\), \(f\) and \(S\) are given functions with values in \([0,\,\infty )\) and \(\mathbb {R}^{3\times 3}\), respectively. Here \(S\) satisfies \(|S(x,n,c)|<S_0(c)(1+n)^{-\alpha }\) with \(\alpha \geq 0\) and some nonnegative nondecreasing function \(S_0\). With the tensor-valued sensitivity \(S\), this system does not possess energy-type functionals which seem to be available only when \(S\) is a scalar function. We can establish a priori estimation to overcome this difficulty and explore a relationship between \(m\) and \(\alpha \), i.e., \(m+\alpha >\frac{7}{6}\), which insures the global existence of bounded weak solution. Our result covers completely and improves the recent result by X. Cao and Y. Wang [Discrete Contin. Dyn. Syst., Ser. B 20, No. 9, 3235–3254 (2015; Zbl 1322.35061)] which asserts, just in the case \(m=1\), the global existence of solutions, but without boundedness, and that by M. Winkler [Calc. Var. Partial Differ. Equ. 54, No. 4, Article ID 922, 3789–3828 (2015; Zbl 1333.35104)] which only involves the case of \(\alpha =0\) and requires the convexity of the domain.