Machine Translation (MT) is usually viewed as a one-shot process that generates the target language equivalent of some source text from scratch. We consider here a more general setting which assumes an initial target sequence, that must be transformed into a valid translation of the source, thereby restoring parallelism between source and target. For this bilingual synchronization task, we consider several architectures (both autoregressive and non-autoregressive) and training regimes, and experiment with multiple practical settings such as simulated interactive MT, translating with Translation Memory (TM) and TM cleaning. Our results suggest that one single generic edit-based system, once fine-tuned, can compare with, or even outperform, dedicated systems specifically trained for these tasks.