This paper addresses the Energy-Aware Distributed Hybrid Flow Shop Scheduling Problem with Multiprocessor Tasks (EADHFSPMT) by considering two objectives simultaneously, i.e., makespan and total energy consumption. It consists of three sub-problems, i.e., job assignment between factories, job sequence in each factory, and machine allocation for each job. We present a mixed inter linear programming model and propose a Novel Multi-Objective Evolutionary Algorithm based on Decomposition (NMOEA/D). We specially design a decoding scheme according to the characteristics of the EADHFSPMT. To initialize a population with certain diversity, four different rules are utilized. Moreover, a cooperative search is designed to produce new solutions based on different types of relationship between any solution and its neighbors. To enhance the quality of solutions, two local intensification operators are implemented according to the problem characteristics. In addition, a dynamic adjustment strategy for weight vectors is designed to balance the diversity and convergence, which can adaptively modify weight vectors according to the distribution of the non-dominated front. Extensive computational experiments are carried out by using a number of benchmark instances, which demonstrate the effectiveness of the above special designs. The statistical comparisons to the existing algorithms also verify the superior performances of the NMOEA/D.