  This paper deals with a boundary-value problem in three-dimensional smooth
bounded convex domains for the coupled chemotaxis-Stokes system with slow
$p$-Laplacian diffusion \begin{equation}\nonumber
  \left\{
  \begin{aligned}
  &n_t+u\cdot\nabla n=\nabla\cdot\left(|\nabla n|^{p-2}\nabla
n\right)-\nabla\cdot(n\nabla c), &x\in\Omega,\ t>0,\ \
  &c_t+u\cdot\nabla c=\Delta c-nc,&x\in\Omega,\ t>0,\ \
  &u_t=\Delta u+\nabla P+n\nabla\phi ,&x\in\Omega,\ t>0,\ \
  &\nabla\cdot u=0, &x\in\Omega,\ t>0,\ \
  \end{aligned}
  \right. \end{equation} where $\phi\in W^{2,\infty}(\Omega)$ is the
gravitational potential. It is proved that global bounded weak solutions exist
whenever $p>\frac{23}{11}$ and the initial data $(n_0,c_0,u_0)$ are
sufficiently regular satisfying $n_0\geq 0$ and $c_0\geq 0$.
