Reflection high-energy electron diffraction (RHEED) intensity oscillations recorded along high-symmetry azimuths during the homoepitaxial growth of GaAs(111)A thin films by molecular-beam epitaxy show reentrant behaviour, with oscillations occurring at high and low temperatures but disappearing at intermediate values over a ~10 o C temperature window. In off-symmetry azimuths, however, the oscillations persist at all temperatures. Scanning tunnelling microscopy images indicate growth-mode changes with temperature consistent with the RHEED results. RHEED patterns and rocking curves provide further evidence for a model based on a transition between two-dimensional layer-by-layer growth and step propagation.