Both the U.S. and Australia allow dual citizenship. Many Americans and Australians are dual citizens, many without knowing it, by reason of lineage and place of birth.
BUT … Australia does not allow members of parliament to be dual citizens.
Until recently, the prohibition of dual citizenship on members of parliament was not well-known. Lack of knowledge of the prohibition combined with lack of knowledge by many Australians of being dual citizens has now resulted in a mess, with members being declared ineligible or at the risk of being declared ineligible, including the Deputy Prime Minister of the country, Barnaby Jones.
Barnaby Jones was born to an Australian mother and a New Zealand father, in Australia. This apparently makes him a natural born dual citizen of both New Zealand and Australia, although he has never formally activated his New Zealand citizenship
Nor have he ever resided in New Zealand, attended New Zealand schools, been a part of New Zealand’s social insurance system, served in the military of New Zealand, voted in New Zealand elections, paid taxes to New Zealand, or in any other informal way activated his New Zealand citizenship.
As another member of parliament is currently in court concerning the application of the prohibition to his peculiar circumstances in being a dual citizen, we will see whether there is any wiggle room for Australians who innocently discover that they are or are eligible for becoming dual citizens.
During last year, the question of dual citizenship arose in two other cases: U.S. Senator Ted Cruz and British Foreign Secretary Boris Johnson.
Ted Cruz was born to a U.S. citizen mother in Canada, where his parents were legal residents, making him a natural born dual citizen of the U.S. and Canada. (Just to be sure, Cruz was not also a natural born citizen of Cuba, the birthplace of his father, as Cuba revokes the citizenship of persons who leave that worker’s paradise.) When the issue of his allegiance to this country arose last year, when he was running for the his party’s nomination for President and the birther’ism bug was going around, Cruz renounced “his” Canadian citizenship.
Boris Johnson was born to British citizen parents in the United States, making him a natural born citizen of the U.K. and the U.S. When the birther’ism bug was particularly virulent, it was pointed out that Johnson was a dual citizen. This prompted the Internal Revenue Service to make inquiry as to taxes owed to the United States, since the U.S. taxes income on a global basis. Boris Johnson responded with the equivalent of a corporate inversion and renounced “his” U.S. citizenship.
In the U.S., a person is a natural born U.S. citizen by reason of any one of three condition:  being the child at birth of a U.S. citizen mother,  being the child at birth of a U.S. citizen father and,  being born in the U.S. (I will not go into further details, of which there are several.)
Some people think “natural born” means all three, or is only patrilineal. While such arguments can be made, practice and law are otherwise, and two U.S. Supreme Court decisions have long decided the matter. Hundreds of decisions by state Secretaries of State and Election Commissioners, acting in a magisterial capacity, and many lower court decisions have affirmed the meaning of natural born.
Nevertheless, from time to time, people entertain themselves with birther’ism. For example, the Washington Post entertained its readers with birther theories when Ted Cruz was a candidate for the Republican nomination for President. In prior years, Barack Obama (a non-U.S. citizen father) and John McCain (born in the Canal Zone) have born the brunt of birther’ism. And, now, the Aussies are having their fun with it.