His features contort with obvious pain as he tells his story, his memories of Caroline clearly something he holds precious.
What makes the bolded section dependent? What's it missing to form a clause. Is it a type of supplemental clause.?
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The dependent clause is not a clause at all. The subject of the phrase is not taking any action. To consider if something is a clause, try to separate it out from the rest of the sentence and see if it makes sense as a sentence. "His memories of Caroline clearly something he holds precious," doesn't make any sense because the phrase contains an apparent subject "memories" but lacks a verb.
Rather, the part after the comma is just a plain old phrase and serves to elaborate on the sentence's primary clause. This construction is relatively uncommon in English. Consider similar sentences like, "He crept, afraid that his pursuers may spot him, to the place he had been told was safe," or, "Fed up with their behavior, she slammed the encyclopedia down on the table." These two sentences use a phrase for elaboration in the same way as your example sentence, though both place the elaborating phrase in different places of the sentence than your example.