Summary: In this paper, we consider the chemotaxis-Navier-Stokes system with nonlinear diffusion and rotational flux given by \[\begin{aligned} n_t +u\cdot\nabla n=\Delta n^m-\nabla\cdot(uS(x,n,c)\cdot\nabla c),\quad & x\in\Omega,\quad t>0,\\ c_t + u\cdot\nabla c=\Delta c-c+n,\quad & x\in\Omega,\quad t>0,\\ u_t + k(u\cdot\nabla)u=\Delta u+\nabla P+n\nabla\phi,\quad & x\in\Omega,\quad t>0\\ \nabla\cdot u=0,\quad & x\in\Omega,\quad t>0\end{aligned}\] in a bounded domain \(\Omega\subset\mathbb{R}^3\), where \(k\in\mathbb{R}\), \(\phi\in W^{2,\infty}(\Omega)\) and the given tensor-valued function \(S\): \(\overline\Omega\times[0,\infty)^2\rightarrow\mathbb{R}^{3\times 3}\) satisfies \(|S(x,n,c)|\leq S_0(n+1)^{-\alpha}\ \text{for all }\ x\in\mathbb{R}^3,\ n\geq 0,\ c\geq 0.\) Imposing no restriction on the size of the initial data, we establish the global existence of a very weak solution while assuming \(m+\alpha>\frac{4}{3}\) and \(m>\frac{1}{3}\).