Sorry, I answered a completely different question. My answer is for multiclass classification, while you asked for multilabel classification. My mistake!
A brute-force way to approach your original problem, multilabel classification, is to transform it into multiclass classification by creating all possible combination of the labels and consider each combination a single class.
In this way you can use the method I explained above, but you have to consider the powerset of the set of all labels, which can be huge.