The Multiscale Hybrid-Mixed (MHM) method for the Stokes--Brinkman equations [1,2] relies on primal finite element discretizations, and therefore the discrete Cauchy stress tensor does not belong to the space $H(div,\Omega)$. In this talk we present results from [6], where we propose an element-by-element reconstruction of the Cauchy stress tensor on the second-level meshes, constructed from the MHM velocity and pressure fields, adapting the approaches introduced in [3,4]. We prove that the reconstructed stress belongs to $H(div,\Omega)$ and achieves optimal convergence rates in the $L^2(\Omega)$-norm. Furthermore, we introduce a new fully computable a posteriori error estimator and establish its reliability and efficiency that, in contrast to the residual-based estimator proposed in [5], is guaranteed. The theoretical results are validated by a series of numerical experiments. REFERENCES [1] R. Araya, C. Harder, A.H. Poza, and F. Valentin, Multiscale Hybrid-Mixed Methods for the Stokes and Brinkman equations - A Priori Analysis, IAM J. Numer. Anal., 63(2):588–618, 2025. [2] R. Araya, C. Harder, A. Poza, and F. Valentin, The multiscale hybrid-mixed method for the Stokes and Brinkman equations - The method, Comput. Methods Appl. Mech. Engrg., 324:29–53, 2017. [3] A. Hannukainen, R. Stenberg, and M. Vohralı́k, A unified framework for a posteriori error estimation for the Stokes problem, Numer. Math., 122(4):725–769, 2012. [4] G.R. Barrenechea, L. Martins, W. Pereira and F. Valentin, An H(div, Ω)-conforming flux reconstruction for the Multiscale Hybrid-Mixed method, (submmited). [5] R. Araya, R. Rebolledo, and F. Valentin, On a multiscale a posteriori error estimator for the Stokes and Brinkman equations, IMA J. Numer. Anal., 41(1):344–380, 2021. [6] L. Martins, R. Rebolledo, and F. Valentin, An H(div, Ω)-conforming post-processing of the Cauchy stress tensor in Multiscale Hybrid-Mixed approximations of the Stokes–Brinkman problem, (in preparation).