  This paper investigates an incompressible chemotaxis-Navier-Stokes system
with slow $p$-Laplacian diffusion \begin{eqnarray} \left\{\begin{array}{lll}
n_t+u\cdot\nabla n=\nabla\cdot(|\nabla n|^{p-2}\nabla
n)-\nabla\cdot(n\chi(c)\nabla c),& x\in\Omega,\ t>0, c_t+u\cdot\nabla c=\Delta
c-nf(c),& x\in\Omega,\ t>0, u_t+(u\cdot\nabla) u=\Delta u+\nabla
P+n\nabla\Phi,& x\in\Omega,\ t>0, \nabla\cdot u=0,& x\in\Omega,\ t>0
\end{array}\right. \end{eqnarray} under homogeneous boundary conditions of
Neumann type for $n$ and $c$, and of Dirichlet type for $u$ in a bounded convex
domain $\Omega\subset \mathbb{R}^3$ with smooth boundary. Here, $\Phi\in
W^{1,\infty}(\Omega)$, $0<\chi\in C^2([0,\infty))$ and $0\leq f\in
C^1([0,\infty))$ with $f(0)=0$. It is proved that if $p>\frac{32}{15}$ and
under appropriate structural assumptions on $f$ and $\chi$, for all
sufficiently smooth initial data $(n_0,c_0,u_0)$ the model possesses at least
one global weak solution.
