This paper discusses the dynamic behavior of the generalized Keller–Segel–Navier–Stokes system in a smoothly bounded convex domain \(\Omega \subset \mathbb {R}^{3}\) marginally governed by \(\begin{aligned} {\left\{ \begin{array}{ll} n_{t}+u\cdot \nabla n=\Delta n-\nabla \cdot (n{{\mathcal {S}(x,n,c)\cdot \nabla c}})+\nabla \cdot (n\nabla \phi ),& \quad x\in \Omega ,t>0,\\ c_{t}+u\cdot \nabla c=\Delta c-c+n,& \quad x\in \Omega ,t>0,\\ u_{t}+\kappa (u\cdot \nabla )u+\nabla P=\Delta u-n\nabla \phi +n{{\mathcal {S}(x,n,c)\cdot \nabla c}},& \quad x\in \Omega ,t>0,\\ \nabla \cdot u=0,& \quad x\in \Omega ,t>0, \end{array}\right. } \end{aligned}\) wherein the gravitational potential \(\phi \in W^{2,\infty }(\Omega )\) , and \(\mathcal {S}(x,n,c)\) represents a given tensor-valued function with saturation satisfying \(\begin{aligned} |\mathcal {S}(x,n,c)|\le C_{\mathcal {S}}(1+n)^{-\alpha }\quad \text{ with }~C_{\mathcal {S}}>0~\text{ and }~\alpha >0. \end{aligned}\) It is readily apparent that for each \(\kappa \in \mathbb {R}\) and for arbitrary sufficiently nonnegative initial data \((n_{0},c_{0},u_{0})\) , then the corresponding no-flux Dirichlet boundary value problem admits at least one globally weak solution provided that an appropriate modest presumption on the parameter \(\alpha \) holds, namely \(\begin{aligned} \alpha >\frac{2}{3}. \end{aligned}\)
