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The Family History Library (FHL) in Salt Lake City Utah has 19th C. civil registration records starting in 1843. Unfortunately they have not been digitized yet, so they are only available on microfilm in Salt Lake, when the FHL reopens. They also have a collection of oral family histories from Papeete, some of which are available on-line.
The original civil registration records were microfilmed in Archives nationales, Section Outre-Mer, Paris.
Dear Mauliola Aspelund,
Thank you for posting your request on History Hub!
In addition to the information from the previous poster, we suggest that you search the French Polynesia, Civil Registration, 1780-1999 FamilySearch index, the Archives Nationales d'Outre-Mer, Etat Civil (French National Archives of Overseas ), and the Récolement archives d'État Civil (French Polynesia Archives Service). In addition, the FamilySearch Research wiki for Society Islands (includes Tahiti) may be useful.
We hope this is helpful. Best of luck with your research!