To me, "ringing" seems like an adjectival participle, describing the state that the alarm bells are put into. However, it does not simply modify "alarm bells", but is like a second argument to the verb "set" in the pattern "set + noun + resulting state".
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Anonymousset alarm bells ringing ... Is "ringing" an adjective functioning as a modifier of the noun phrase "alarm bells" or a verbal noun, a head of the phrase "alarm bells ringing"?No. Definitely not. That would be "ringing alarm bells", i.e., "alarm bells that are ringing". If they were already ringing, there would be no need to set them ringing!