The Cloaking Device was an unseen device used by Christine Johnson to conceal an Anomaly's presence.
Christine Johnson somehow concealed an Anomaly at her Headquarters using the cloaking device so her organisation could perform missions to the future. One day, when Christine wanted to terminate a Future Predator, the cloaking device had to be turned off and the Anomaly Research Centre detected the Anomaly. It was later reactivated.
James Lester later speculated that Christine had some sort of cloaking device when he had cause to suspect she was still hiding an Anomaly at her headquarters.
It is possible that the cloaking device used the same phenomenon that occurred in Episode 1.4: the thick metal doors of a kitchen store blocked the magnetic effects of that anomaly. It is supported by the fact that the room that Christine's Anomaly was in had metal chains draped over all the walls, although it could be disproved because the cloaking device was momentarily turned off and the Anomaly Detection Device used radio interference to detect Anomalies, not magnetic interference.