In subduction zones, stress is released by earthquakes and transient aseismic slip. The latter falls into two categories: slow slip and afterslip. Slow-slip events emerge spontaneously during the interseismic phase, and show a progressive acceleration of slip with a negligible contribution of synchronous tremors or microseismicity to the energy, or moment release. In contrast, afterslip occurs immediately after large and moderate earthquakes, decelerates over time, and releases between 20 and 400% of the moment released by the preceding earthquake. Here we use seismic and GPS data to identify transient aseismic slip that does not fit into either of these categories. We document a seismic-aseismic slip sequence which occurred at shallow depths along a weakly coupled part of the Andean subduction zone in northern Peru and lasted seven months. The sequence generated several moderate earthquakes that together account for about 25% of the total moment released during the full sequence, equivalent to magnitude 6.7. Transient slip immediately followed two of the earthquakes, with slip slowing at a logarithmic rate. Considered separately, the moment released by transient slip following the second earthquake was more than 1,000% of the moment released during the earthquake itself, a value incompatible with classical models of afterslip. Synchronous seismic swarms and aseismic slip may therefore define a stress-release process that is distinct from slow-slip events and afterslip.