This answer may not be timely, but I think this should solve your problem.
Try adding both tags to both ex sentences. I did this, and it worked for me in a similar case.
q=mainquery&fq=status:public&fq={!tag=dt}doctype:pdf&fq={!tag=doc}document:1&facet=on&facet.field={!ex=dt,doc}doctype&facet.field={!ex=dt,doc}document
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