That’s why I said some 
It is in Bermuda at least.
Right, I checked on the issue of Prime Minister Mackenzie and additional senators, and I’m pleased to find that my aging memory was working reasonably correctly.
After Mackenzie won the general election in 1874, following becoming Prime Minister in 1873, the Conservatives held a majority in the Senate. Mackenzie was concerned that he wouldn’t get his legislation through the Senate, so he requested the appointment of additional senators.
However, the British government advised Her Majesty not to agree, because in their view, the appointment of additional senators was only to be used in extraordinary circumstances, when the Senate was actually blocking legislation passed by the Commons, not routinely every time a new government came to power without a majority in the Senate. Since Mackenzie’s concern was general in nature, not tied to any particular difficulty in getting legislation passed, the British government did not think the power to appoint extra senators should be used.
Since at this time, the British government could advise Her Majesty on matters relating to Canada, Her Majesty of course followed their advice and declined to direct the GovGen to appoint additional senators.
(Source: McConnell, Commentary on the British North America Ac)
Bermuda isn’t an independent Commonwealth realm like Canada, Australia, or Jamaica; it’s a British overseas territory, like Gibraltar or the British Virgin Islands. Its Governor (not a Governor General) is appointed on the advice of the British government, not a local one.
(Nevertheless, according to Wikipedia, it apparently has the oldest continuously functioning parliament in the world.)
Canadians can travel visa-free to the U.S. in almost all immigration categories, which can be very handy in a time crunch. The same applied to Canadian landed immigrants who were citizens of Commonwealth countries until a few years ago, but not anymore. Sadly, I can’t find a decent cite at the moment; all the links on the US State Department website seem to be broken.
ETA: here’s a cite.
UDS and I had a discussion on this several years back, in this thread: When Did Canada Stop Being Part of the English Monarchy?. We couldn’t agree on the answer, so we left it as one of those unsolved mysteries that will have to wait until it actually occurs to see what happens. 
There are at least two different ways to analyse the question. One is the approach you suggest here: that the constitutions of the various Commonwealth realms do not themselves define who is the monarch, leaving that to the British to define, so if the British change the succession, that automatically changes the succession in all the Commonwealth realms as well.
Textual support for that analysis can be found in provision such as the Preamble to Canada’s Constitution Act, 1867, which provides:
The rest of the Act simply refers to “the Queen”. Taken together, these provisions suggest that whoever is the monarch of the United Kingdom is automatically the monarch of Canada.
However, the opposing view is that while that approach was accurate when the Constitution Act, 1867 was originally enacted, and Canada was a self-governing dominion but not independent of the U.K., it has since been modified by subsequent constitutional developments, notably the patriation of the Constitution of Canada in 1982. That was done by the Canada Act 1982. The key provision for this discussion is s. 2, which reads:
So the counter-argument is that since the U.K. Parliament no longer has the power to extend its laws to Canada, a British Act of Parliament that changes the succession law for the purposes of British law does not affect the succession law for Canada.
Not without legislation to enact it. The Canadian Parliament must pass exactly an similar Bill to the British Bill that changes it.
That’s exactly the point in debate. Do you have a cite for that assertion? (I agree with you, by the way, but wonder if you could assist by providing authority.)
Well, when Edward VIII abdicated, all the self-governing British Dominions passed legislation through their legislatures to mirror the change of monarch. (The Irish Free State refused to do so for a while so, de jure, it had a different monarch to everyone else for a short time)
There’s been a number of references to the Australian constitutional crisis of 1975. In the present day, what is the chance of that action (a gov. general removing a PM) being repeated? There doesn’t seem to be any strictly legal reason it couldn’t happen, but would the political concensus allow it? And what would happen in other Commonwealth nations if the gov. general decided to act that way?
This is speaking not from a strict legal standpoint but as a bit of informed opinion: It could happen, in the sense that the power inheres in the Crown (i.e., in the GG acting in behalf of the Queen), but is there on a “Break glass only in case of Emergencies” basis. That is, in a constitutional crisis where the Government is in fact exceeding its proper authority, it’s there as a “nuclear option” if needed. But the odds that it will actually ever need to be used are probably miniscule.
We had a similar situation in late 2008. In response to legislation that would have basically bankrupted them, the three opposition parties agreed that the Liberals and the NDP would form a coalition government. In response, before a vote of no confidence could be held, Prime Minister Harper asked the Governor General to prorogue Parliament (suspend it, ending the session).
On one hand, it was clear that the Prime Minister’s command of the confidence of the House was questionable at best, even if that hadn’t yet been demonstrated by a vote. On the other hand, it would be very unprecedented, and quite questionable, for the Governor General to refuse a Prime Minister’s request, especially since, by convention, he would have had to resign.
In the end, she granted his request. The effects of this decision on constitutional convention have yet to become clear.
Here the Dismissal was often characterised in terms of Australian antagonism to the Monarchy (although in truth, it was more antagonism to Kerr). Only minor amendments to the Australian Constitution were made as a result of it and the Governor-General retains the reserve power to dismiss the PM.
Whitlam had nominated Kerr to the post in the first place, and if he failed to secure a compliant yes-man who would do his bidding he had only himself to blame.
Don’t all death penalty cases in Jamaica have their final appeal in the Privy Council? I think it is still true that the Privy Council has some power in independent Commonwealth nations.
She also sometimes flies this flag when abroad in places where there’s no native royal standard for her: File:Personal flag of Queen Elizabeth II.svg - Wikipedia
In addition to all this head of state stuff, Commonwealth citizens also get other benefits, in very limited cases: if born or adopted prior to 1983 to a UK citizen mother, they have the right of abode in the UK (essentially, automatic citizenship) via their mothers, which ordinarily would only pass from a UK-citizen father.
So, if you were born in Kenya in 1980 to a Kenyan man and a British woman, you can be a UK citizen. If you were born in Algeria in 1980 to an Algerian man and a British woman, you can’t (except through the ordinary naturalization channels), although you can be a French citizen instead in that particular case.
Visiting a country where your country doesn’t have consular representation, and lost your passport? No worries - you can go to the British embassy, which will issue you one.
You can even stand for office in Britain! Well, you have to register a permanent address there, so technically you have to move there first.
Well that seems sexist
It does, doesn’t it? Still, prior to 1983, just being born in the UK (with the same exceptions as under current US law) got you British citizenship. Plus, they went the opposite way from then until 2006: unmarried British fathers could not pass on British citizenship, unless the birth occurred in countries which treat children born to unmarried parents the same as children born to married parents for the purposes of their own citizenship laws.
ETA: That is to say, if I (as a UK citizen) had knocked up some girl in France in 2004, the resulting crotchfruit would not have been entitled to UK citizenship unless I married the girl.
Apparently, as a result of this and the constitutionally less extreme but politically contentious prorogation a year later, the current search for a replacement for Madame Jean, when her term ends later this year, has placed constitutional knowledge high on the list of desirable characteristics.
And for the non-Canadians, the Canadian equivalent of the Australian GG v. PM constitutional crisis mentioned earlier is the 1925 King-Byng Affair.