A metric gauge on a set is a maximal collection of metrics on the set such that the identity map between any two metrics from the collection is locally bi-Lipschitz. We characterize metric gauges that are locally branched Euclidean and discuss an obstruction to removing the branching. Our characterization is a mixture of analysis, geometry, and topology with an argument of Yu. Reshetnyak to produce the branched coordinates for the gauge.
"On the locally branched Euclidean metric gauge." Duke Math. J. 114 (1) 15 - 41, 15 July 2002. https://doi.org/10.1215/S0012-7094-02-11412-4