  We consider an initial-boundary value problem for the incompressible
four-component Keller-Segel-Navier-Stokes system with rotational flux
$$\left\{\begin{array}{l} n_t+u\cdot\nabla n=\Delta
n-\nabla\cdot(nS(x,n,c)\nabla c)-nm,\quad x\in \Omega, t>0,\\ c_t+u\cdot\nabla
c=\Delta c-c+m,\quad x\in \Omega, t>0,\\ m_t+u\cdot\nabla m=\Delta m-nm,\quad
x\in \Omega, t>0,\\ u_t+\kappa(u \cdot \nabla)u+\nabla P=\Delta u+(n+m)\nabla
\phi,\quad x\in \Omega, t>0,\\ \nabla\cdot u=0,\quad x\in \Omega, t>0
\end{array}\right.$$ in a bounded domain $\Omega\subset \mathbb{R}^3$ with
smooth boundary, where $\kappa\in \mathbb{R}$ is given constant, $S$ is a
matrix-valued sensitivity satisfying $|S(x,n,c)|\leq C_S(1+n)^{-\alpha}$ with
some $C_S> 0$ and $\alpha\geq 0$. As the case $\kappa = 0$ (with
$\alpha\geq\frac{1}{3}$ or the initial data satisfy a certain smallness
condition) has been considered in [14], based on new gradient-like functional
inequality, it is shown in the present paper that the corresponding
initial-boundary problem with $\kappa \neq 0$ admits at least one global weak
solution if $\alpha>0$. To the best of our knowledge, this is the first
analytical work for the {\bf full three-dimensional four-component}
chemotaxis-Navier-Stokes system.
