Can a participle phrase contain a subordinate clause?

I can't find any sources that explain whether a participle phrase can contain a subordinate clause.

Is that really possible?

For example...

- The ambulance arrived to transport a man struggling to remain conscious while bleeding from the nose.

In this example above, the subordinate clause "while bleeding from the nose" seems to be embedded in the participle pharse "struggling to remain conscious", because it is nonsensical that the ambulance is bleeding, but I'm not sure whether English permits this kind of sentence.