  We analyse a nonlinear stochastic partial differential equation that
corresponds to a viscous shallow water equation (of the Camassa--Holm type)
perturbed by a convective, position-dependent noise term. We establish the
existence of weak solutions in $H^m$ ($m\in\mathbb{N}$) using Galerkin
approximations and the stochastic compactness method. We derive a series of a
priori estimates that combine a model-specific energy law with non-standard
regularity estimates. We make systematic use of a stochastic Gronwall
inequality and also stopping time techniques. The proof of convergence to a
solution argues via tightness of the laws of the Galerkin solutions, and
Skorokhod--Jakubowski a.s. representations of random variables in quasi-Polish
spaces. The spatially dependent noise function constitutes a complication
throughout the analysis, repeatedly giving rise to nonlinear terms that
"balance" the martingale part of the equation against the second-order
Stratonovich-to-It\^{o} correction term. Finally, via pathwise uniqueness, we
conclude that the constructed solutions are probabilistically strong. The
uniqueness proof is based on a finite-dimensional It\^{o} formula and a
DiPerna--Lions type regularisation procedure, where the regularisation errors
are controlled by first and second order commutators.
