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Multi-modal fusion approaches aim to integrate information from different data sources. Unlike natural datasets, such as in audio-visual applications, where samples consist of "paired" modalities, data in healthcare is often collected asynchronously. Hence, requiring the presence of all modalities for a given sample is not realistic for clinical tasks and significantly limits the size of the dataset during training. In this paper, we propose MedFuse, a conceptually simple yet promising LSTM-based fusion module that can accommodate uni-modal as well as multi-modal input. We evaluate the fusion method and introduce new benchmark results for in-hospital mortality prediction and phenotype classification, using clinical time-series data in the MIMIC-IV dataset and corresponding chest X-ray images in MIMIC-CXR. Compared to more complex multi-modal fusion strategies, MedFuse provides a performance improvement by a large margin on the fully paired test set. It also remains robust across the partially paired test set containing samples with missing chest X-ray images. We release our code for reproducibility and to enable the evaluation of competing models in the future.