If someone living at an embassy abroad gave birth to a child on the embassy grounds, then what country and what place will be indicated on the birth certificate?
To clarify the question, let's assume that we mean the US embassy in another country. But I'm also interested in how a similar situation will occur in the embassies of other countries.
I saw a similar question but it is related to citizenship, but I only ask about birth certificate
The place of birth on the birth certificate is where the child was actually born. Indeed, usually it will be more specific than city and state or province and will also identify a hospital or residence or other place where the birth happened.
So, for example, if a child is born to U.S. diplomats in Paris, France (in or out of the embassy grounds), the birth certificate will say that the child was born in Paris, France at Charles de Gaulle Hospital.
But, that child will still be a U.S. citizen in all likelihood, because that child's mother, and/or married father or unmarried father who acknowledges paternity, is a U.S. citizen (in all likelihood) pursuant to 8 U.S.C. §§ 1401 and 1409. The child may or may not be a French dual citizen depending upon the citizenship law of France.
In the case of a French diplomat who has a child born physically in Washington D.C. (inside or outside the French embassy) the birth certificate will likewise state that the child was born in Washington D.C.
The French diplomat's child, however, will not be a U.S. citizen since Section 1 of the 14th Amendment to the U.S. Constitution's first sentence (which is also found in 8 U.S.C. § 1401(a)) states:
All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside.
But a French diplomat's child is not "subject to the jurisdiction" of the United States, so the French diplomat's child does not gain U.S. citizenship at birth (assuming for simplicity's sake that both of the child's parents are French citizens and are not U.S. citizens) despite the fact that the child was born in the United States.
If we assume that the ambassador and his wife (or other embassy staff), living in the embassy residence, will give birth to a child, then what country and what place will be indicated on the birth certificate?
There is a common misconception that embassies are the sovereign territory of the country whose embassy it is. This is not the case. Embassies are inviolable, meaning that the country where the embassy is located can't exercise physical jurisdiction over the premises of the embassy without the ambassador's consent. This is established by article 22 of the Vienna Convention on Diplomatic Relations. The premises otherwise remain legally under the sovereignty of the country where the embassy is located.
For example, if someone commits a crime in the French embassy to the US, the prosecution would take place in the DC courts according to the DC criminal code (or, depending on the nature of the crime, in the federal courts according to the federal criminal code, which is rather more likely for a crime committed in an embassy than it is for a crime committed somewhere else).
In other words, the grounds of the French embassy in Washington are US territory and the grounds of the US embassy in Paris are French territory. Therefore, a birth certificate for a child born at the French embassy to the US will be issued by the District of Columbia, and it will show that the child was born in the District of Columbia at 4101 Reservoir Road NW.
This assumes that the birth is reported; if everyone in attendance at the birth is a diplomat with full immunity then they could choose to ignore the requirement to report the birth. Presumably they would only do that if they had a way of certifying the birth under French law. I imagine that some countries have provisions for their capital cities to issue birth certificates for children of their diplomats born abroad, but I'd be a bit surprised if France were among them.
A child's place of birth is the place where they are born (obviously). Embassies have no special status, embassies in France are all in France, and French embassies are in many different countries. Born in a Greek embassy in France, your place of birth is France.
Sometimes your place of birth is legally important, but then there is a good chance that there are specific rules for that case. For example the baby of German diplomats born in the German embassy in Washington is born in the USA, but because of the special case this baby doesn't get the rights and duties that 99.9% of children born in the USA have, and would most likely instead get all the rights and duties that 99.9% of children in Germany have.
(Practically, I don't know whether they would get some special entry in their passports so for example US border officers wouldn't keep asking "but your place in birth is the USA, you must be US citizen. And if such a baby lived in the USA for 18 years, I wonder if they would have the right to naturalise. Same as any other 18 year old, or more rights because they were born in the USA. And now we can go totally hypothetically and ask if such a baby could eventually become US president, but I think the rule there is something like "born as US citizen" and not actually "born in the USA". And if a golf club accepted only members "born in the USA", which might be illegal discrimination or not, they'd have to accept that German child of German diplomats born in the USA).
