When a U.S. Citizen gives birth to a child outside the U.S., that child is automatically considered a U.S. Citizen once they are brought back to the U.S. as an immigrant. The parents of the child can request a Certificate of Citizenship or U.S. Passport for the child if desired, but no further paperwork is necessary. However, this is only for those children who are going to permanently live in the U.S.
For children born to at least one U.S. Citizen who are going to live abroad, the parents must apply for naturalization of the child. This can only happen to children whose U.S. Citizenship parent had been living in the U;S; for at least 5 years prior to the child's birth and at least two of those years were after the age of 14. During the naturalization process, the child must live temporarily in the U.S.
Once the child is born, the parents must register the birth at the nearest U.S. Embassy as soon as possible. This will make establishing U.S. Citizenship easier.
When bringing the child back to the United States, you would need to apply for the child's immigrant Visa at the U.S. Embassy.
I am NOT a legal authority, but it sounds like you need to get some legal advice. The situation depends on where the child resides, why the father was in Tahiti, was there a marriage, and possibly other factors. As an example, I was born in a foreign country where my Dad was assigned (US Army), my Mom was with him, and I was brought back as a dual citizen when I was 3 mo. old on my mother's (newly updated) passport. Possibly their is an embassy or a legal counsel that you can ask. Good Luck!!
I was born in Tahiti, my mom is from Tahiti and my dad is from the US, so I have dual citizenship.