We present a method to minimize distortion and compute globally injective mappings from non-injective initialization. Many approaches for distortion minimization subject to injectivity constraints require an injective initialization and feasible intermediate states. However, it is often the case that injective initializers are not readily available, and many distortion energies of interest have barrier terms that stall global progress. The alternating direction method of multipliers (ADMM) has recently gained traction in graphics due to its efficiency and generality. In this work we explore how to endow ADMM with global injectivity while retaining the ability to traverse non-injective iterates. We develop an iterated coupled-solver approach that evolves two solution states in tandem. Our primary solver rapidly drives down energy to a nearly injective state using a dynamic set of efficiently enforceable inversion and overlap constraints. Then, a secondary solver corrects the state, herding the solution closer to feasibility. The resulting method not only compares well to previous work, but can also resolve overlap with free boundaries.