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A generalized approach to decompose the compressible Navier-Stokes equations into an equivalent set of coupled equations for flow and acoustics is introduced. As a significant extension to standard hydrodynamic/acoustic splitting methods, the approach provides the essential coupling terms, which account for the feedback from the acoustics to the flow. A unique simplified version of the split equation system with feedback is derived that conforms to the compressible Navier-Stokes equations in the subsonic flow regime, where the feedback reduces to one additional term in the flow momentum equation. Subsonic simulations are conducted for flow-acoustic feedback cases using a scale-resolving run-time coupled hierarchical Cartesian mesh solver, which operates with different explicit time step sizes for incompressible flow and acoustics. The first simulation case focuses on the tone of a generic flute. With the major flow-acoustic feedback term included, the simulation yields the tone characteristics in agreement with reference results from Kühnelt based on Lattice-Boltzmann simulation. On the contrary, the standard hybrid hydrodynamic/acoustic method with the feedback-term switched off lacks the proper tone. As the second simulation case, a thick plate in a duct is studied at various low Mach numbers around the Parker-beta-mode resonance. The simulations reveal the flow-acoustic feedback mechanism in very good agreement with experimental data of Welsh et al. Simulations and theoretical considerations reveal that the feedback term does not reduce the stable convective flow based time step size of the flow equations.