I have read the basics of how force based physics simulation works and how you can create joints by enforcing certain constraints on the forces between two rigid bodies. What is unclear to me is what happens during the integration step. 

In particular even if you enforce the constraints on the forces - during the integration step the bodies will still move out of alignment. By this I mean the hinge will seperate. What is the next step to address this?

How do you translationally re-align the bodies in a way which is consistent with classical mechanics? If you choose one piece as the "ground truth" and basically re-align all the connected pieces to it, you are biasing the evolution of the system to be around that piece which can't be correct.