The Fox News veteran took heat online after many said he should have come to his colleague's aid rather than ask his own question.
A almost identical exchange took place in January 2017, when then president-elect Trump refused Acosta's question, saying: "Not you - not you - you're fake news".
Trump then called on Fox News correspondent John Roberts and called Fox "real news". "To issue a blanket condemnation of the network as "fake news" is ... unfair".
Later in the press conference, Trump identified Sun journalists in the crowd, saying that he wishes they put the positive things he said about May in the headline.
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Interestingly, the man who heads "real network" Fox News, Rupert Murdoch, also owns the "fake news" Sun. Welker was cut off when she tried to reply. "Lets go to a real network".
"Asking smart, tough questions, whether in a presidential press conference or interview, is central to the role a free press plays in a healthy republic", she said.
The network's news site, FoxNews.com, avoided mentioning Trump's labeling of the Sun interview in its account of the news conference. Co-host Joe Scarborough frequently broke in while Trump talked on Thursday, calling some of his claims untrue. There was no indication whether the live fact-check was considered a failed experiment, or whether it would be repeated on "Morning Joe" or any other show.
In the article, Trump criticized May's handling of Brexit, saying that she would likely "kill" any trade deals between the US and the United Kingdom because she didn't take his advice.
The Sun also recorded the interview and inserted audio clips into the article.
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"I said very good things about her".
Later in the press conference, Trump slammed reporter Kristen Welker after a question about Russian Federation. I didn't think they'd put it in, but that's alright.
"To say the President called us "fake news" with any serious intent is, well... fake news", the Sun concluded. "You see, that is such dishonest reporting", Trump said.
Mason took it off and asked his question.
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