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\nContext. We present V<sc>ENICE</sc>, an operator-splitting algorithm to integrate a numerical model on a hierarchy of timescales.\nAims. V<sc>ENICE</sc> allows a wide variety of different physical processes operating on different scales to be coupled on individual and adaptive time-steps. It therewith mediates the development of complex multi-scale and multi-physics simulation environments with a wide variety of independent components.\nMethods. The coupling between various physical models and scales is dynamic, and realised through (Strang) operators splitting using adaptive time-steps.\nResults. We demonstrate the functionality and performance of this algorithm using astrophysical models of a stellar cluster, first coupling gravitational dynamics and stellar evolution, then coupling internal gravitational dynamics with dynamics within a galactic background potential, and finally combining these models while also introducing dwarf galaxy-like perturbers. These tests show numerical convergence for decreasing coupling timescales, demonstrate how V<sc>ENICE</sc> can improve the performance of a simulation by shortening coupling timescales when appropriate, and provide a case study of how V<sc>ENICE</sc> can be used to gradually build up and tune a complex multi-physics model. Although the examples provided here couple dedicated numerical models, V<sc>ENICE</sc> can also be used to efficiently solve systems of stiff differential equations.\n