If a baby is born on an international flight over international waters, what nationality are they?

In vast majority of the cases, the child will have one or more of the parents' nationalities through jus sanguinis (this is true even if the child is born in some country). As far as I know, every country in the world (except the Vatican where nationality is ex officio) has some sort of jus sanguinis system, where children born abroad to parents of that nationality also have that nationality, under some circumstances. Some countries impose certain conditions that need to be met for the parent to transmit nationality to children born abroad, e.g. the parent needs to have resided in the country for a certain number of years, or the parent cannot have themselves gotten nationality by descent, etc. So depending on the circumstances, there's a small chance that a parent won't be able to transmit nationality via jus sanguinis. There's an even smaller chance that both parents won't be able to transmit it.

In the tiny chance that the child does not get a nationality through jus sanguinis, it could be that the law of the country of the airplane's registration provides its nationality to a child who would otherwise be stateless born on its airplane. All countries that are a party to the Convention on the Reduction of Statelessness of 1961 must provide this; but not many countries are party to the convention. A few other countries also provide this.

In the tiny chance that the above all don't apply, the child will be born stateless. However, it is still likely that one or more of the parents' countries of nationality will have a procedure to get residency or nationality for the child after birth upon the parent's application.

Only countries that recognise jus soli ("right of the soil") consider babies born within their borders to be citizens of that country. Today, only countries in the Americas and a handful of others recognise jus soli. So, for the majority of the world it's not relevant where a baby is actually born.

However, wherever a baby is born there is generally a birth certificate of some sort issued by the local government. A birth certificate does not confer the right of citizenship.

Airlines obviously do not issue birth certificates, so the first opportunity to issue one would be on the ground at the destination (or wherever the plane actually lands, if it is considered an emergency).

If one or both of the parents are from a country which operate a jus sanguinis system, the baby would be entitled to the citizenship of that country. The vast majority of countries have a jus sanguinis system, either by itself or in conjunction with a jus soli system (which entitles the baby to citizenship if s/he is born on that country's soil).

If, however, the baby is not so entitled to citizenship, the question gets a bit more complicated, and I can't find any answer which I trust:

• It seems to be widely repeated on the Internet that

  The United Nations considers a child born in-flight to have been born in the airplane's registered country. (source)

  However, I can't find a UN document that suggests that this is in fact the opinion of the UN (admittedly, I haven't looked very hard). Nevertheless, if the operator's registered country did agree with this interpretation, and they run a jus soli system, then the child will have that citizenship.

• Wikipedia suggests that the coordinates at the time of birth will be compared to the origin and destination, and look at the closer country's to see if they operate a jus soli system. This strikes me as somewhat impractical, since I'm not sure anyone would take care to note down the coordinates of the plane in the middle of an unexpected birth. In addition, are you comparing the airports, or merely which country is closer? What happens if the plane diverts (as would likely occur during a birth)?

• In practice, I have a feeling that the answer is it depends. Since most countries operate a jus sanguinis system, and there aren't that many in-flight births, it probably isn't that clear. What you really want to avoid is whatever country being decided on as the place of birth operating only jus sanguinis, with both parents only having the citizenship of countries that operate only jus soli.