Dense branching morphologies (DBM) obtained in thin gap electrodeposition cells are characterized by a dense array of branches behind a flat advancing envelope. In this Letter, we show the existence in DBM of a new (porous) phase, qualitatively different from a (compact) metal deposit. The local porosity inside the branches is found to be much more robust than geometric characteristics such as the width or the distance between branches. This fact seems to be unreported in previous modeling of DBM. A mean-field model is proposed that displays overall features observed in the experiments, such as concentration profiles, front velocity, and branched internal structure.