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.