  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$^\circ$ and 84$^\circ$. In contrast to air
showers with more vertical incidence, these inclined air showers illuminate
large ground areas of several km$^2$ 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 radio signals grows with the zenith angle of the air shower. Inclined air
showers with EeV energies are thus measurable with sparse radio-antenna arrays
with grid sizes of a km or more. This is particularly attractive as radio
detection provides direct access to the energy in the electromagnetic cascade
of an air shower, which in case of inclined air showers is not accessible by
arrays of particle detectors on the ground.
