Which goes to the nub of the question of any Australian Republic having a directly elected president, when inevitably there comes to be a political discord between the elected Head of State and the elected Head of Government.
They act independently. This letter from the Queen’s Private Secretary after the 1975 Australian crisis spells it out:
As we understand the situation here, the Australian Constitution firmly places the prerogative powers of the Crown in the hands of the Governor-General as the representative of the Queen of Australia. The only person competent to commission an Australian Prime Minister is the Governor-General, and The Queen has no part in the decisions which the Governor-General must take in accordance with the Constitution. Her Majesty, as Queen of Australia, is watching events in Canberra with close interest and attention, but it would not be proper for her to intervene in person in matters which are so clearly placed within the jurisdiction of the Governor-General by the Constitution Act.
oops double post, sorry
A couple of points: Though several people have alluded to it and Northern Piper stated it briefly, it’s important to note that the Queen of the United Kingdom is not the Queen of Canada or Australia. That is, the Duke of Edinburgh’s wife is in fact the person who holds those titles, and they converge in her by law, but they’re separate and distinct legal entities. In theory Her Majesty the Queen of Canada might declare war on herself as Her Majesty the Queen of the Bahamas – though one part of the Commonwealth system is to make this only an over-the-top hypothetical theoretical example. By way of parallel, think of Attorney Brown, who is a natural person at law, but has a distinct identity as managing trustee of the Smith Family Trust, and while he can sue and be sued in this role, if exercising proper diligence in the role he cannot be held responsible as a natural individual for any debts the Trust may incur. In fact, as likewise trustee for the Jones Family Trust, he may find it incumbent on himself in that capacity to institute suit against himself as trustee of the Smith trust. What the Queen is conceived of as doing as Queen of Canada is legally distinct from what she may do as Queen of the U.K. or of Australia.
Second, in Canada at least there are a small but significant range of powers reserved to her in propria persona and not able to be exercised by the Governor General. The most obvious one is the naming of a new Governor General (effectively always on the advice of the incumbent Government, but still her duty and power as Queen). In fact, an example of where this might come into play is in the Tom Clancyish scenario of terrorists detonating a pony nuke at the State Opening of Parliament in Ottawa, when Governor General, Government, and effectively every Senator and M.P. are present. The Queen might well ring up Halifax and say to the Lt. Gov. of Nova Scotia, “You’re interim Gov. Gen. Give orders to apprehend those terrorists and get a Government in place.” – and it would be completely legal, her doing her job as Sovereign in an unprecedented situation.
Coomonwealth in a Nutshell:
“Help, get me out of this nutshell!”
I’ve come to prefer monarchies for their honesty. We might as well have an inbred parasite chosen through accident of birth, because - as it turns out - it makes not the slightest bit of difference anyway. I like the monarchy because it reminds us that real power is a club from which almost everybody is excluded.
It ain’t theory. Its fact. Its happened. In 1948, the King of India was at war with the King of Pakistan for more than a year. George VI.
Awkward*
But it took him until 1952 to eliminate himself? 
Seriously, I think one if not both of them were republics by the time they went to war, though I may be wrong.
The most important powers are the reserve powers: powers which the GovGen can exercise in extraordinary circumstances, to keep the constitutional government operating. For example, just last year, the Prime Minister of Canada asked the Gov Gen to prorogue Parliament (i.e. - end the ongoing session) during a parliamentary crisis. Eventually, Her Excellency decided to take that advice, but there are some who argue she should have refused, because the reason that the PM wanted the prorogation because there was a possibility that the government might fall to a non-confidence vote.
More dramatically, in Australia, the GovGen in the 1970’s dismissed the Prime Minister and appointed the Leader of the Opposition as Prime Minister. The GovGen’s justification for doing so was the dismissed PM had not been able to get his budget through Parliament, and was allegedly trying to borrow money from the international markets to keep the government going without a budget. If so, that was a gross breach of the principles of parliamentary democracy and responsible government. The GovGen of the time thought that it was a serious enough breach to exercise his reserve power to dismiss the PM. Thirty years on, the propriety of his decision is still debated.
It becomes necessary at this point to highlight one falsehood about the monarchy.
Walter Bagehot famously wrote, “But the Queen has no such veto. She must sign her own death-warrant if the two Houses unanimously send it up to her.” While this over-the-top ne plus ultra of Parliamentary power was in fact true when Bagehot wrote it in 1848, the situation changed in 1931 with passage of the Statute of Westminster.
Now for the British Parliament to pass such a measure would be seen as attempted regicide of the Queen of Canada, the Queen of New Zealand, etc. Instead, [the UK] Parliament would need to unanimously adopt a joint resolution calling on the Parliaments of HM’s other realms to severally and unanimously adopt the said royal death-warrant. Only when presented with a warrant concurred in by all her realms would the Queen be required to sign it into law.
One trusts that George V and his successors have breathed more freely knowing this.
Another monir iten: The Queen has a personal flag for some Commonwealth Realms, which is flown in her presence when she’s there. This flag takes precedence over the national flag. Here’s her personal Canadian flag flying over the Peace Tower in Ottawa on Canada Day a few days ago.
Yes. The Crowns are legally completely separate in each Commonwealth Realm. It’s as if, in Canada, the Constitution started out with, “We’re a monarchy, and we’re going to adopt the British Royal Family as monarchs.” Theoretically, Canada could decide on a different succession from Queen Elizabeth II, or a different monarch entirely… or none at all.
Personally, I think it’s brilliant. We have a Royal Family, but we only have to pay for it when they’re actually here, and most of the time, they’re in other countries. 
And Queen Elizabeth II has been at war with herself as well. During the invasion of Grenada (a Commonwealth Realm), forces from other Commonwealth Realms in the Caribbean participated, though the main invasion force was US.
Well, it’s not like the GGs and LGs don’t incur expenses. The GG in Canada had a budget of 42 million in 06-07 (no comment on how that compares to the British monarchy expenses)
Presumably, we’d still be paying for a head of state and heads of provinces no matter what form they took, unless we wanted to roll their offices into those of the appropriate heads of government. Which I think would be a bad idea.
Well if you’re paying for a head of state anyway, it’s not a positive to have an absentee monarch as one.
Nope, India Republic 1950, Pakistan after much debate 1956.
I know other people have commented on the distinction between the Crown of the UK and the Crown of Australia, but nobody’s commented on precisely this. Australia is independent of the UK in all manners, in name as well. Both countries have separate monarchs (though the title is held by the same physical person), and separate governments that don’t have any influence on each other. If the diplomatic links between Australia and the UK are closer than in other similar countries (and I don’t know if they are), it’s because both countries decide so because of their shared history.
Am I wrong or is the duty to temporarily increase the size of the Senate when advised to do so by the Prime Minister, as done under Brian Mulroney in the 1990s to pass the GST bill, for some reason reserved to the Monarch itself and not the Governor General? If so, I wonder how this came to be.
Lise Thibault, former Lieutenant-Governor of Quebec, has been under scrutiny for sending forged expense reports while in function, and I believe that criminal charges have now been filed against her.
Okay - now I get it. Thank you for the explanation.
What would those reasons be? Or would it be about the shared bonds as described by other posters?
Now, if Australia did decide to become a Republic, would the response by the others in the Commonwealth be:
-Traitor! We hate you forever!
or
-Meh. Okey-dokey then.
?