Staying more than 90 days in Schengen area as a French citizen, but only have a US passport

Legally speaking, as a French citizen you can stay in France indefinitely. This right does not depend on the possession of a document showing that you are French.

Similarly, you can stay in other Schengen countries under the rules set forth in the EU’s free movement directive. For stays of up to three months in duration (in each country), you only need to be able to show a valid identity card or passport. Now the free movement directive does not say that this passport must be issued by an EU country, but you can imagine that asserting your free movement rights as a French citizen in (for example) Germany by showing a US passport is not going to get you very far. They’ll want some evidence that the French government recognizes you as a French citizen.

Once you’re in the Schengen area, you’re very unlikely to be asked to prove that you’re there legally, and even if you are asked after you’ve been there longer than 90 days, there’s a good chance you’ll be able to get away with showing your US passport and talking your way out of any negative consequences by claiming French nationality. But maybe you won’t be able to, and you are almost certain to be delayed and possibly fined if you try to leave the Schengen area showing only a US passport after more than 90 days.

Contrary to the assertion in another answer, it’s not necessary to "switch your arrival record" if you enter on a US passport and then acquire French documents in France. Your right of residence in France and in the Schengen area does not depend on having entered with a French document. As long as you can prove you’re French, you can benefit from these rights.

If your parents didn’t register your birth to the French civil register, you must petition for a French nationality certificate from the French Nationality Office in Paris if you reside outside of France (which I assume from the wording of your question).

Only then are you eligable to recieve a French ID/Passport and to be treated as an EU citizen.

A birth certificate issued in the US is not, in itsself, proof of French citizenship.

It can only assist in verifying French citizenship if at least one of the parents stated in the certificate are also French citizens.

Collecting the information about your parents (their french passport, ID, last address in France etc.) would be the first step.

Check if they actually registered you in the French civil register the second step (see point 2 of the french link below).

If not, start the application process with the Paris Court (service of French nationality), which I believe can be time consuming (see point 5 of the french link below)

You email or file your application (completed, dated, signed and supporting documents) Paris Court (service of French nationality).

• Tribunal judiciaire de Paris – Paris – 75 – Annuaire | Service-public.fr

Sources:

• French citizenship by birth abroad to at least one French citizen – Wikipedia.
• Identity Card / Passport: how to prove your french citizenship? | Service-public.fr

…
2. Check your birth certificate
If you were born in Foreigner [outside of France]
If you do not have a recent identity document, proof of French nationality can be provided by presenting a birth certificate (full copy or extract with filiation) of less than 3 months.

This act of civil status can establish your French nationality if you are in one of the following cases:

• The birth certificate (on secure paper) was issued by the Nantes Central Registry Office.
• The birth certificate (on plain paper) was issued by an embassy consular vital statistics officer.

…
5. Request a certificate of French nationality
If you do not have any of the above documents, you will need to request a French nationality certificate (CNF).

Regardless of the legality it will be infinitely easier for you to travel if you have a French passport. I suggest contacting your local consulate with whatever documentation you have (birth certificate of your and preferably proof of citizenship of your parents).

You can certainly enter France with a US passport and try to sort it out locally while you are in France. Just make sure you carry all documentation that you may need.

Provided you meet all the criteria to be a French citizen, it’s probably not illegal for you to stay for more than 90 days, but if you get checked after 90 days with only your US passport, you will have a LOT of explaining to do. Passport checks at Schengen borders are rare but they do happen occasionally and you will have to show it when you leave Schengen. Hotels may also want to see your passport when you check in.

Recommendation: sort it your before you travel.

There are a number of ways to prove French citizenship, including:

• an ID card or passport
• a birth certificate (subject to certain conditions)
• a certificate of French nationality

Source: https://www.service-public.fr/particuliers/vosdroits/F18713?lang=en