The global athletics watchdog, which brought the initial case against the 22-year-old, said on Twitter that it would challenge the decision at the Court of Arbitration for Sport (CAS).
The AIU had alleged that Eid Naser, who was born in Nigeria and competes for Bahrain, missed three out-of-competition tests within a one-year period, which would constitute an anti-doping violation and carries a minimum one-year ban.
However, the World Athletics disciplinary panel cleared Eid Naser last month because on one of the tests, in April 2019, the anti-doping inspector became confused over the numbering of the flats in her building and rang the wrong door bell.
The panel added that the case was "very much on the borderline".
Under the so-called whereabouts rule, elite athletes must make themselves available for random out-of-competition testing and state a location and one-hour window where they can be found on any given day. (Reporting by Brian Homewood, editing by Ed Osmond)