Results and plans to apply interferometry to air shower observations at the Pierre Auger Observatory

By analysing the radio emissions from air showers using interferometry, we can estimate their properties. In this contribution, we apply interferometry to reconstruct air-shower parameters based on measurements taken with the Auger Engineering Radio Array (AERA) at the Pierre Auger Observatory. This reconstruction method is achievable at AERA through precise clock synchronisation with a beacon and an accurate survey of the station locations. Interferometry has been applied to several thousand inclined air-shower observations for the first time, which allows for tests on the performance of air-shower geometry reconstruction, recovery of the radio signal from low-energy air showers, and methods to study the polarisation of the radio-emission mechanisms. Additionally, in this contribution, we will also provide an overview of efforts to enable interferometry for the recently installed radio detectors that are part of the AugerPrime upgrade.