The sentence structure requires a dichotomy. "Real" is correctly opposed to "fake" although now that I think about it "genuine" would be a better choice than "real"."Neither true nor false" is a different concept. I conceive "fake" news as  stories that have the appearance of news but are not intended to be believed. "False" news is stories story that have the appearance of news and are intended to be believed but are not true. 

To take a historical example, the stories about Iraqi WMDs were false news, not fake news. The Onion publishes fake news, not false news.