Given a differential manifold with smooth metric g, does there exist an orthonormal coframe $e^a$ such that $d^*e^a$=0
a. everywhere or almost everywhere
b. locally/globally, and
c. is it unique?
Here, $d^*$ denotes the codifferential (https://en.wikipedia.org/wiki/Hodge_star_operator#Codifferential). The closest reference I could find is a physics reference related to gauge fixing tetrad fields (https://arxiv.org/pdf/1603.07571.pdf) and does not appear to address the above existence and uniqueness question.
A reference addressing the questions would be appreciated.