Conducting Legally Efficient Administration and Resolution of Patents Act of 2021 or the CLEAR Patents Act of 2021 This bill establishes that a federal agency proceeding pertaining to a patent shall, in certain instances, be stayed while certain administrative validity challenges to that patent are pending. Specifically, if the U.S. Patent and Trademark Office institutes an inter partes review (IPR) or a postgrant review (PGR) of a patent and the cancellation of one of the patent's claims would materially affect a federal agency proceeding, that proceeding must be stayed until the IPR or PGR has been decided or otherwise terminated.