From what I have read about Australian immigration policy, Trump’s telephone tantrum with Australian Prime Minister Turnbull makes no sense. Trump should be mad with Obama, not Turnbull. This USA Today article gives some background.
Australia has been refusing to accept refugees in a way Trump should like. Australian was being inundated with refugees arriving by boat, many of them from the Middle East. Australia refused to accept the boats and sent the boat people to islands belonging to Papua New Guinea, where living conditions were poor. Refugee activists criticized Australia harshly for this policy.
The details are not clear to me, but it appears that Obama agreed to take some of the refugees that Australia refused to take. Trump presumably thinks some of the refugees that the US agreed to take are bad people from the Middle East who were trying to escape to Australia.
Another article that shed some light on the deal was an op-ed by Roger Cohen in the New York Times. As a Jew writing for the Trump-hating New York Times, Cohen has to make fun of Trump for putting Rex Tillerson at State in such a bad position, but he at least describes it better than most reports about it. Cohen says the deal was signed in September, but kept secret until after the election, because refugees were such a sensitive issue in the campaign. He says the Australians told the American negotiators, “We really want to mothball these places,” the island camps, because they had become an acute embarrassment to Australia. Cohen says he has visited one of the camps to see how bad they were.
It looks as if Obama was just trying to do Australia a favor, to help Australia close the camps and get rid of some of the bad publicity that they created for Canberra. Ironically, Australia had refused to take the same refugees that Trump was refusing to take. The two men agreed on refugee policy, but Trump apparently thought he was getting stuck with the worst of the deal. Presumably, Australia is still refusing to accept the refugees and is trying to get the US to carry out Obama’s promise to take some of them. But now Trump doesn’t want them either.