The islands with a flourishing 'third gender'

Thank you. The distinction between formal authority and informal power is an interesting model, but I would disagree that patriarchy is then a correct term to use since patriarchy does not distinguish between formal and informal in any definition that I have read. Matriarchy and patriarchy is usually defined by whom holds the primary power in a society. One can't say that women are hold more power in society and then say its a patriarchy, unless we define patriarchy in different terms to that commonly used. It also looks a bit like switching the definition based on needs, like how a monarch with a queen as supreme formal authority is still not a matriarchy because the overall informal power is held by men.

I do have a major question however about that list of powers. If a dispute between two families is about marriage, do the male chieftains hold authority to decide on the matter? My guess would be that they do not as that would cross the boundary of power. It sounds more like this society have male gender role that give them official domain over war and trade with other tribes, and a female gender role that give them domain over all other areas. This would be very similar to many other tribal societies, with the rare exception that allow individuals to choose which one of the two roles a person want. Many people object to this classification since it would mean that most historical societies are neither matriarchy or patriarchy but rather structured around gender roles where power is held supreme with in each gender group.

> In that situation most of the formal power was held by men. The members of parliament, the generals, the treasurer, the postmaster, the judges -- all were men.

The time when the queens power were supreme the parliament did not hold power over the monarch, nor did the generals, treasurer, postmasters or judges. Supreme power is supreme. The UK parliament was created on the idea to limit the power of the monarch, same with the treasurer which reported only to the parliament, but those rules took ages to be put in place.

> Having power doesn't mean they don't have to take into account the informal power around them.

There is a very strong issue of definition here. A supreme judge have to take into account the facts of a case but they still hold exclusive power to make a judgement. That is the question here: do the chiefs hold the exclusive power to decide the case or are they not allowed to do so when it cross over to the domain that women decide on.

> When a society is labeled patriarchal it is not, in practice, a statement about all the power in that society.

That is why the term patriarchal and matriarchy is about which group holds primary power. Not all, but most power.

> that women would be the presidents and generals and judges and men would basically chill at home and smoke the fish and do the knitting and not (overtly) control public policy. No one has ever found a society like this.

This is just a different way of defining formal power to be more meaningful than informal. We could role reversal a society where women held power in all trading, war and negotiation with foreign tribes and men held power over all other domains. Both men and women would do cooking, fishing and knitting, but men would control domestic policy while women controlled diplomatic policy.

Is that a matriarchy? In my view no. The primary power in society is not about who control the generals and who is allowed to go and fight wars with neighboring tribes. There is more to tribal society than war.