This paper proposes an adaptive energy efficient task allocation scheme for a multiprocessor system-on-a-chip (SoC) in real-time energy harvesting systems. The proposed scheme generates an energy efficient offline task schedule for a multiprocessor SoC energy harvesting system by balancing application workload among multiple processing elements and pushing real-time application towards their deadlines. The offline task schedule is dynamically extended to adapt to the energy availability in the runtime to improve the probability of a task to be feasibly scheduled. Simulation experiments show that the proposed scheme achieves energy savings of up to 24%, and reduces task deadline miss ratio of up to 10%.