<p>A recurring challenge in science and engineering is the model–reality gap, where trusted legacy simulators lose fidelity due to unresolved physics or structural incompleteness. This challenge has motivated remedies ranging from imperfect mechanistic models to fully data-driven surrogates. Here, we address this gap with Alternating Neural Integrators (ANI), a non-intrusive reuse-and-correct framework for upgrading executable legacy simulators without requiring access to or modification of their internal implementation. Guided by operator-splitting principles, ANI alternates the evolution of a fixed prior simulator with a learned neural correction that targets structured discrepancy between the prior and the supervisory data. We show that ANI can recover missing coupling in chaotic systems and act as an effective subgrid correction in turbulence, improving dynamical fidelity where prior models drift. In selected mechanistically structured settings, post hoc symbolic distillation yields compact hypotheses and, in controlled benchmarks, supports closed-loop refinement of the prior. By combining data-driven flexibility with reusable scientific simulators in a theoretically grounded framework, this work provides a practical gray-box route for systematically upgrading existing computational infrastructure when a callable prior and supervisory data are available.</p>