I heard intimations that the Canadian government might sue the British government over the submarines we recently bought from them.
Presumably it would look something like Her Majesty the Queen in Right of Canada vs. Her Majesty the Queen in Right of the United Kingdom.
Are there any potential complications involved in one of the Queen’s legal persons suing another?
It’s not the Queen that doing the suing: it’s the Crown. Within Australia, states and the Commonwealth of Australia (the federal government) can sue each other: a quick search in a legal database turned up Commonwealth v Western Australia (1999) 196 CLR 392 and South Australia v Commonwealth (1962) 108 CLR 130. The first case would be the Crown in right of the Commonwealth of Australia suing the Crown in right of the State of Western Australia.
What Giles said is on target. The crown of X is the sovereign acting in her capacity as monarch and protector of realm X, according to the political uses of X – which may have legal disputes with the crown of Y, which is her acting in her capacity as monarch and protector of realm Y. English Dopers-at-Law would have to fill in the details, but I believe there was a trade dispute between Great Britain and Hanover while one of the early Georges was king of both, way antedating any of the sovereignty-of-nations-in-the-Empire/Commonwealth concept.
It might be pointed out that this legal-person thing is not limited to sovereigns – it is quite possible for an individual to sue himself in his other legal persona as a LLC where an “actual case or controversy” can exist. In fact, one of the New Deal laws was invalidated as unconstitutional by the Supreme Court owing to a lawsuit in which a man who was president and majority stockholder of a company filed suit in his individual capacity against the company which he owned and ran, to prevent himself-as-a-company from doing what that law required, because it would reduce the income which himself-as-a-stockholder would receive from himself-as-a-company. And SCOTUS held that that was a legitimate case or controversy – he as company president was expected to comply with constitutional laws; he as stockholder was entitled to the maximum income he could legitimately receive from his investment in the company, so he was placed in an untenable situation which the courts were entitled to resolve, by considering his (as stockholder) claim that the law he (as company president) was obliged to follow was unconstitutional.
There’s two separate aspects to your question, matt - one substantive, and the other procedural.
The substantive question is whether the Crown in right of one Commonwealth realm can sue the Crown in right of another Commonwealth realm. That’s not a problem. At one time, the constitutional position was that the Crown was indivisible, so such suits would pose a theoretical problem - how could the same entity sue itself? However, as part of the evolution from Empire to Commonwealth, it’s become recognised that the Crown is divisible, and such lawsuits are therefore possible.
For more information on this point, you might want to check out this thread from last February: When did Canada stop being part of the English monarchy?
Here’s what I posted on this point at that time:
A related aspect of the substantive question is whether there is a defence of Crown immunity. I wouldn’t think that would be the case here, where the dispute would essentially be one of contract law. At common law, the Crown could not be sued in its own courts. That has been changed by statute, and most Commonwealth jurisdictions that have the Crown as head of state have a statute permitting such suits. For example, at the federal level in Canada there is the Crown Liability and Proceedings Act, which states that Her Majesty in right of Canada is legally liable in the same way as a natural person (subject to some exceptions). (See Part I of the federal Act, dealing with liability.) There is a similar U.K. statute governing the liability of the Queen in right of the U.K., but I’ve not been able to find it on-line. So I would think that the Crown in right of the U.K. could potentially be sued under the contract of lease and sale.
The procedural question is in which court, and under what style of cause, the Crown in right of the United Kingdom can be sued. The Crown Liability Acts normally deal with this point as well. See section 21 of the federal Act, which provides that the Crown in right of Canada can be sued in any provincial superior court, or the Federal Court. I would assume that the equivalent British legislation would confer similar jurisdiction on the High Court of England and Wales, and likely on the Scottish courts.
Finally, there’s the style of cause. There’s considerable variation here. The federal statute states that suits against the Crown in right of Canada are to be brought against the Attorney General of Canada (see s. 23). Other jurisdictions may give different directions in their Crown proceedings statutes. For example, some Canadian provinces provide that the style of cause should be against the “Government of [Province’s Name].” That’s not universal - sometimes suits are brought in the name of Her Majesty in right of the Province. See, for example, The Queen in the Right of the Province of Ontario v. Board of Transport Commissioners, [1968] S.C.R. 118 and Her Majesty in right of the Province of Alberta v. Canadian Transport Commission, [1978] 1 S.C.R. 61.
The style of cause of the potential lawsuit would thus depend on the terms of the British Crown Proceedings Act. Does it say that the U.K. government is to be sued under the name of “Her Majesty the Queen”? or by a suit against the minister responsible, as a representative of the government? And since the Canadian federal statute doesn’t apply to the British courts, would it be acceptable for the suit to be brought by the Attorney General of Canada, or would it have to be brought by “Her Majesty the Queen in right of Canada”?
In the event of a lawsuit, there will likely be various articling students working on these neat little points of procedure.
As always, this is not intended as legal advice. If you are planning on bringing a lawsuit against the British government, please consult a solicitor familiar with British law. 
I work for the Australian Post Office. I’m in a large sprawling building surrounded by a carpark and a truck parking yard. The Australian Postal Corporation owns the building, the carpark, and half of the truck yard. The other half of thar yard it rents from itself.