Weak solutions to the 2D or 3D stochastic NSCHEs

We consider a diffuse interface model for the mixture of two incompressible fluids driven by transport noise. Under suitable abstract assumptions, we prove the existence of a global weak martingale solution as well as the pathwise uniqueness of a global strong solution in the two dimensional case. This is the first result addressing a unified framework derived from the works by Brze{ź}niak & Motyl and Mikulevicius & Rozovskii, on the study of stochastic partial differential equations.