The Biot system is widely used to model poro-elasticity problems, where the deformation of an elastic porous medium is coupled with the flow of a Newtonian fluid. Besides the displacement and the pore pressure fields, two other variables play a major role in simulations: the stress tensor and the average fluid (Darcy) velocity. In this work, we propose a post-processing technique to recover those dual variables from a previously computed Primal Hybrid solution for the Biot system. In the Primal Hybrid method [1], the H1-conformity of the primal variables is relaxed through the introduction of Lagrange multipliers, which can be linked to the normal component of the dual variables. This, in turn, allows for a local H(div)-conforming recovery of the dual variables on the whole domain. By adapting the ideas from [2] to the Biot context, we present a post-processing strategy to obtain well-balanced approximations for both the velocity and stress fields. The resulting approximations show good conservation/equilibrium properties, similar to those verified by the mixed-mixed formulations for Biot system [3]. In the present work, we prove such conservation results and show a draft for the convergence analysis. Numerical illustrations are also presented, showcasing the post-processing technique.