Mesoscale eddies are ubiquitous features of the global ocean circulation. Traditionally, anticyclonic eddies are thought to be associated with positive temperature anomalies while cyclonic eddies are associated with negative temperature anomalies. However, our recent study found that about one-fifth of the eddies identified from global satellite observations are cold-core anticyclonic eddies (CAEs) and warm-core cyclonic eddies (WCEs). Here we show that in the tropical oceans where the probabilities of CAEs and WCEs are high, there are significantly more CAEs and WCEs in summer than in winter. We conduct a suite of idealized numerical model experiments initialized with composite eddy structures obtained from Argo profiles as well as a heat budget analysis. The results highlight the key role of relative wind-stress-induced Ekman pumping, surface mixed layer depth, and vertical entrainment in the formation and seasonal cycle of these unconventional eddies. The relative wind stress is found to be particularly effective in converting conventional eddies into CAEs or WCEs when the surface mixed layer is shallow. The abundance of CAEs and WCEs in the global ocean calls for further research on this topic.