To simultaneously translate a source language into multiple different target languages is one of the most common scenarios of multilingual translation. However, existing methods cannot make full use of translation model information during decoding, such as intra-lingual and inter-lingual future information, and therefore may suffer from some issues like the unbalanced outputs. In this paper, we present a new approach for synchronous interactive multilingual neural machine translation (SimNMT), which predicts each target language output simultaneously and interactively using historical and future information of all target languages. Specifically, we first propose a synchronous cross-interactive decoder in which generation of each target output does not only depend on its generated sequences, but also relies on its future information, as well as history and future contexts of other target languages. Then, we present a new interactive multilingual beam search algorithm that enables synchronous interactive decoding of all target languages in a single model. We take two target languages as an example to illustrate and evaluate the proposed SimNMT model on IWSLT datasets. The experimental results demonstrate that our method achieves significant improvements over several advanced NMT and MNMT models.