We develop a pathwise theory for scalar conservation laws with spatially dependent fluxes and quasilinear multiplicative ''rough path'' dependence, a special case being spatially dependent stochastic conservation laws with quasilinear stochastic dependence. Following some of the ideas of a previous note where we considered spatially independent fluxes, we introduce the notion of pathwise stochastic entropy solutions, which is closed with respect to the local uniform limits of paths, and prove that, in the presence of a single ''rough path'', it is well posed, that is we establish existence, uniqueness and continuous dependence in the form of (pathwise) $L^1$-contraction. Our approach is motivated by the theory of stochastic viscosity solutions, which was introduced and developed by two of the authors, to study fully nonlinear first- and second-order stochastic pde with multiplicative noise. This theory relies on special test functions constructed by inverting locally the flow of the stochastic characteristics. For conservation laws this is best implemented at the level of the kinetic formulation which we follow here.