Bairstow was stumped by Carey after he had left the crease without the ball being called dead.
An entertaining final day came to an end with Australia clinching the victory by 43 runs after a sensational Ben Stokes century, as England tried to chase 371 runs.
However, apart from Stokes’ century, the focus was on the way Bairstow was stumped by Alex Carey.
After dodging the last ball of the over, Bairstow marked his crease and moved to the other end leaving his crease, but Alex Carey's throw into the stumps caught him beyond the line. The decision went to the TV umpire who deemed that Bairstow was out, which rankled many England players and supporters.
England coach Brendon McCullum echoed the sentiment and said that his team was not going to fraternize with the Australian team anytime soon.
"I can't imagine we'll be having a beer with them any time soon. We have three tests to land some blows and try to win the Ashes. That is where our focus will be,” McCullum told BBC.
The former New Zealand keeper invoked "the spirit of the game" and called it vital.
"In the end, you've got to live with the decisions you make, and that's life. But I feel from our point of view, if we were in the same situation, we might've made a different decision," McCullum stressed after the game,” he added.
Soon after, the Australian media heavily criticized the former New Zealand player, pointing out that McCullum had run out Paul Collingwood in a similar way during the 2009 Champions Trophy. But then Kiwi captain Daniel Vettori called the batter back.