In the task of multi-label classification in data streams, instances arriving in real time need to be associated with multiple labels simultaneously. Various methods based on the k Nearest Neighbors algorithm have been proposed to address this task. However, these methods face limitations when dealing with imbalanced data streams, a problem that has received limited attention in existing works. To approach this gap, this work introduces the Imbalance-Robust Multi-Label Self-Adjusting kNN (IRMLSAkNN), designed to tackle multi-label imbalanced data streams. IRMLSAkNN is based on MLSAkNN and incorporates dynamic thresholds using the Mean Imbalance Ratio per Instance for its punitive removal mechanism and employs the Geometric Mean as the evaluation measure for the self-adjusting window. These adaptations significantly enhance the algorithm's robustness in handling imbalanced data streams. We conducted extensive experiments on 32 benchmark data streams, evaluating IRMLSAkNN against four state-of-the-art kNN-based algorithms using common accuracy-aware and imbalance-aware measures. The results demonstrate that IRMLSAkNN outperforms its competitors in different ranges of imbalance. Additionally, IRMLSAkNN stands out for its time efficiency, making it a competitive solution for the multi-label data stream classification task.