  In this paper we consider the stochastic primitive equation for geophysical
flows subject to transport noise and turbulent pressure. Admitting very rough
noise terms, the global existence and uniqueness of solutions to this
stochastic partial differential equation are proven using stochastic maximal
$L^2$-regularity, the theory of critical spaces for stochastic evolution
equations, and global a priori bounds. Compared to other results in this
direction, we do not need any smallness assumption on the transport noise which
acts directly on the velocity field and we also allow rougher noise terms. The
adaptation to Stratonovich type noise and, more generally, to variable
viscosity and/or conductivity are discussed as well.
