With the Auger Engineering Radio Array (AERA) of the Pierre Auger Observatory, we have observed the radio emission from 561 extensive air showers with zenith angles between 60 and 84. In contrast to air showers with more vertical incidence, these inclined air showers illuminate large ground areas of several km2 with radio signals detectable in the 30 to 80 MHz band. A comparison of the measured radio-signal amplitudes with Monte Carlo simulations of a subset of 50 events for which we reconstruct the energy using the Auger surface detector shows agreement within the uncertainties of the current analysis. As expected for forward-beamed radio emission undergoing no significant absorption or scattering in the atmosphere, the area illuminated by...