In this paper, it is explained how algorithms for convex mixed-integer nonlinear programming (MINLP) based on poly-hedral outer approximation (POA) can be integrated with mixed-integer programming (MIP) solvers through callbacks and lazy constraints. Through this integration, a new approach utilizing a single branching tree is obtained which reduces the overhead required when rebuilding the branching tree in the MIP solver due to the continuous addition of linear constraints approximating the nonlinear feasible region of the MINLP problem. The result is an efficient strategy for implementing a POA utilized by the Supporting Hyperplane Optimization Toolkit (SHOT) solver.