An anomalous climatic event in the tropical Atlantic, starting in autumn 1995 and continuing into the autumn of 1997, is investigated. Using remotely sensed datasets and a reduced-gravity model, it is shown that this event contained both a warm and a cold phase, each of roughly 12 months duration. The propagating signal within each phase is identified in the sea surface height and temperature signatures as both equatorial and off-equatorial waves in the ocean, with coupling to atmospheric convection. The strength, geographical pattern and timing of the anomalous period is consistent with it being the first observed example of a coupled mode of interaction between the ocean and atmosphere found in two previous coupled ocean-atmosphere models of the tropics. This interpretation means that the two-phased climate event, which we will call the Equatorial Atlantic Oscillation (EAO), was associated with purely internal atmospheric and oceanographic variability within the Atlantic basin and hence was independent of the Pacific climate. It can also be inferred, from long-term monthly sea surface temperature and sea-level pressure datasets, that there were potentially several previous EAO events during the past century.