In this work, the flux-coordinate independent (FCI) approach to plasma turbulence simulations is formulated for the case of generic, static magnetic fields, including those possessing stochastic field lines. It is then demonstrated that FCI is applicable to nonlinear turbulent problems with and without X-point geometry. In particular, by means of simulations with the FENICIA code, it is shown that the standard features of ion temperature gradient (ITG) modes are recovered with reduced toroidal resolution. Finally, the impact of ITG turbulent transport on the temperature profile is studied under the influence of a static island, with ITER-like parameters. Results show the wide range of applicability of the method.