Events are inter-related in documents. Motivated by the one-sense-per-discourse theory, we hypothesize that a participant tends to play consistent roles across multiple events in the same document. However recent work on document-level event argument extraction models each individual event in isolation and therefore causes inconsistency among extracted arguments across events, which will further cause discrepancy for downstream applications such as event knowledge base population, question answering, and hypothesis generation. In this work, we formulate event argument consistency as the constraints from event-event relations under the document-level setting. To improve consistency we introduce the Event-Aware Argument Extraction (EA2E) model with augmented context for training and inference. Experiment results on WIKIEVENTS and ACE2005 datasets demonstrate the effectiveness of EA2E compared to baseline methods.