A local hidden variable theory (LHV) of quantum mechanics (QM) assumes that all natural processes are local: information and correlations are propagated at most at the speed of light. The hidden aspect ascribes variability of experimental results to uncontrollable parameters of the model.
There are many local LHV theories that bypass Bell's inequality (and related inequalities) due to loopholes in the measurement process. The most well known loophole is that of detection efficiency.
This site will list those models.