Based on the very recent work by Censor-Gibali-Reich (http://arxiv.org/abs/1009.3780), we propose an extension of their new variational problem (Split Variational Inequality Problem) to monotone variational inclusions. Relying on the Krasnosel'skii-Mann Theorem for averaged operators, we analyze an algorithm for solving new split monotone inclusions under weaker conditions. Our weak convergence results improve and develop previously discussed Split Variational Inequality Problems, feasibility problems and related problems and algorithms.