The use of regularization methods to represent discontinuities and jumps in the solution is appealing for the development of multiphase and multiscale numerical methods, as it allows techniques originally developed for single-phase flows to be easily extended to situations in which variables and their derivatives are not necessarily continuous. The one-fluid method, in which discontinuities in material properties and solution jumps are smeared over a thin layer of controlled thickness $\Delta$, is a representative example. These approaches have become very popular in both scientific and industrial communities due to their simplicity, robustness, and their ability to converge to the exact solution of the discontinuous problem. However, they are known to suffer from reduced accuracy, particularly near interfaces or discontinuities, where derivatives are often contaminated by numerical artifacts resulting from the regularization process. In this presentation, we introduce a new approach for the arbitrary-order reconstruction of discontinuous solutions obtained using regularized methods. We show that, for linear elliptic problems, solutions of regularized formulations typically converge only at first order to the exact discontinuous solution. Remarkably, this apparent limitation can be overcome through an a posteriori sharp solution reconstruction [Fuster & Mimoh, JCP, 2025, Fuster & Sultan, JCP, 2025]. By combining a multiscale framework with asymptotic analysis, the solution of the regularized problem can be corrected to accurately represent discontinuous solutions, including jumps in primitive variables and their fluxes, without modifying the numerical method used to compute the original regularized solution. This approach opens the door to a new class of numerical methods that provide a dual regularized/sharp representation of interfaces. In addition, it offers a natural framework for incorporating subgrid physical models of phenomena occurring along lines, interfaces, or thin regions.