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The physical fidelity of turbulence models can benefit from a partial resolution of fluctuations, but doing so often comes with an increase in computational cost. To explore this trade-off in the context of wall-bounded flows, this paper introduces a framework for turbulence-resolving integral simulations (TRIS) with the goal of efficiently resolving the largest motions using a two-dimensional, three-component representation of the flow defined by instantaneous wall-normal integrals of velocity and pressure. Self-sustaining turbulence with qualitatively realistic large-scale structures is demonstrated for TRIS on an open-channel (half-channel) flow configuration using moment-of-momentum integral equations derived from Navier–Stokes with relatively simple closure approximations. Evidence from direct numerical simulations (DNS) suggests that TRIS can theoretically resolve$$35\,\%{-}40\,\%$$of the turbulent skin friction enhancement for friction Reynolds numbers between$$180$$and$$5200$$, without a noticeable decrease or increase as a function of Reynolds number. The current implementation of TRIS can match this resolution while simulating one flow through time in$${\sim}1$$minute on a single processor, even for very large Reynolds numbers. The framework facilitates a detailed apples-to-apples comparison of predicted statistics against data from DNS. Comparisons at friction Reynolds numbers of$$395$$and$$590$$show that TRIS generates a relatively accurate representation of the flow, while highlighting discrepancies that demonstrate a need for improving the closure models. The present results for open-channel flow represent a proof of concept for TRIS as a new approach for wall-bounded turbulence modelling, motivating extension to more general flow configurations such as boundary layers on immersed objects.