Although machine translation (MT) traditionally pursues “human-oriented” objectives, humans are not the only possible consumers of MT output. For instance, when automatic translations are used to feed downstream Natural Language Processing (NLP) components in cross-lingual settings, they should ideally pursue “machine-oriented” objectives that maximize the performance of these components. Tebbifakhr et al. (2019) recently proposed a reinforcement learning approach to adapt a generic neural MT(NMT) system by exploiting the reward from a downstream sentiment classifier. But what if the downstream NLP tasks to serve are more than one? How to avoid the costs of adapting and maintaining one dedicated NMT system for each task? We address this problem by proposing a multi-task approach to machine-oriented NMT adaptation, which is capable to serve multiple downstream tasks with a single system. Through experiments with Spanish and Italian data covering three different tasks, we show that our approach can outperform a generic NMT system, and compete with single-task models in most of the settings.