It's not really a silly question, it confuses me too I think a good rule of thumb is to agree the verb with the preceeding noun. So either "another example is birds" or "birds are another example", as you suggest. I don't know whether it is technically the case, but I guess people read the first noun first, and therefore it sounds odd when a subsequent verb does not agree; whereas if the verb does not agree with the word after it, we don't mind as much because it sounds natural with the word before it.
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AnonymousAnother example of this are birds.I would change it to