  This paper considers the Keller-Segel model coupled to stochastic
Navier-Stokes equations (KS-SNS, for short), which describes the dynamics of
oxygen and bacteria densities evolving within a stochastically forced 2D
incompressible viscous flow. Our main goal is to investigate the existence and
uniqueness of global solutions (strong in the probabilistic sense and weak in
the PDE sense) to the KS-SNS system. A novel approximate KS-SNS system with
proper regularization and cut-off operators in $H^s(\mathbb{R}^2)$ is
introduced, and the existence of approximate solution is proved by some a
priori uniform bounds and a careful analysis on the approximation scheme. Under
appropriate assumptions, two types of stochastic entropy-energy inequalities
that seem to be new in their forms are derived, which together with the
Prohorov theorem and Jakubowski-Skorokhod theorem enables us to show that the
sequence of approximate solutions converges to a global martingale weak
solution. In addition, when $\chi(\cdot)\equiv \textrm{const.}>0$, we prove
that the solution is pathwise unique, and hence by the Yamada-Wantanabe theorem
that the KS-SNS system admits a unique global pathwise weak solution.
