How to tackle a multilabel classification problem

When each object can be classified from 0 to multiple categories, it is a multilabel classification problem. There are several approachs to tackle this, the most known is probably the One-vs-the-Rest strategy : it consists in dividing the problem into a multitude of binary classification tasks, for each possible label.

However, deep neural networks support inherently multilabel classification. Each neuron of the final classification layer can be associated with a label. As you would like to have multiple output neurons with high values if an object has several tags, you should use a sigmoid activation function on the final layer and the binary crossentropy loss function.

There are several tutorials on the web.

LSTMs, like any other neural net, implicitly support multi-label classification. You should ensure that your output layer has $n$ neurons, one for each class, and you should use logistic activation rather than softmax activation (which is typically used for the final layer in multi-class problems).

Each neuron $N_i$ in the final layer will output a value between 0 and 1. If the output of $N_i$ is greater than 0.5, then the example is a member of label $i$.

Here's a pretty good article that might help.