  We investigate a stochastic version of the Allen-Cahn-Navier-Stokes system in
a smooth two- or three-dimensional domain with random initial data. The system
consists of a Navier-Stokes equation coupled with a convective Allen-Cahn
equation, with two independent sources of randomness given by general
multiplicative-type Wiener noises. In particular, the Allen-Cahn equation is
characterized by a singular potential of logarithmic type as prescribed by the
classical thermodynamical derivation of the model. The problem is endowed with
a no-slip boundary condition for the (volume averaged) velocity field, as well
as a homogeneous Neumann condition for the order parameter. We first prove the
existence of analytically weak martingale solutions in two and three spatial
dimensions. Then, in two dimensions, we also estabilish pathwise uniqueness and
the existence of a unique probabilistically-strong solution. Eventually, by
exploiting a suitable generalisation of the classical De Rham theorem to
stochastic processes, existence and uniqueness of a pressure is also shown.
