Marine soundscapes commonly contain concurrent acoustic events and identifying all components is required to understand interactions between sound classes, for example between sonar and marine mammal vocalisations. Machine learning techniques previously applied to such data primarily aimed to associate a single event type to a specific time interval. Here, we demonstrate how multi-label classification (MLC) systems can automatically identify overlapping (polyphonic) sound combinations. We develop a methodology that uses an input representation previously shown effective for marine sound classification, together with a curated dataset with balanced class distributions and various combinations of sound events in the training data. Using this framework, we test four supervised MLC systems, which yielded comparable performance, with Binary Relevance and Neural Discriminative Dimensionality Reduction multi-task learning systems showing marginal advantages. System performances were affected by the number of overlapping sound classes, the representation of combinations in the training dataset and the complexity or interactions between sound classes. Across systems, vessel noise and delphinid clicks were most accurately classified, while high energy sonar signals, particularly from high frequency echosounders, were frequently misclassified as delphinid tonals. These findings highlight the importance of system design and dataset structure when developing multi-label classification models for marine soundscapes.