In a sentence, an adjective can qualify as an adjective phrase. In grammar, phrases — namely adjective, noun, verb, adverb and prepositional phrases — do not contain both a subject and a verb. They’re smaller than that. A phrase works as the noun or the verb, or one of those three other things.
So if “talking” is basically an adjective phrase in “Mr. Ed was a talking horse,” how do we understand “talking” in, “I saw the horse talking to a cow”? Is it a phrase or a clause?
I asked Geoffrey Pullum, linguistics professor at the University of Edinburgh. He explained that there is no right answer. Such a unit can be either, depending on your interpretation.