Sequence labeling with partially known labels
You can train a regular sequence labeling model (typically CRF) where one of the features is the rule-based predicted label: its value is the actual label when known or a special unknown value otherwise. Given that the model can take into account dependencies between labels (as specified in the parameters) and that the rule-based feature always gives the label except if unknown, the model should be able learn: 


in which cases it should "trust" the rule-based feature (when value is not unknown)
to predict the missing labels by exploiting both the features and the known labels.


Note that in some rare cases the model might still predict a wrong label even if the rule-based label is provided. That would happen if the probability of the sequence is maximized in this way, but it's unlikely to happen if the training data is representative of the distribution.