score:5
As far as I can tell from Article 355(1) of the consolidated TEU/TFEU, French Guiana is part of the EU:
- The provisions of the Treaties shall apply to Guadeloupe, French Guiana, Martinique, RΓ©union, Saint-BarthΓ©lemy, Saint-Martin, the Azores, Madeira and the Canary Islands in accordance with Article 349.
Free movement therefore applies, and you could have entered with your ID card instead of a passport. If that is correct, then the gendarmerie needs better training, since one element of free movement is a policy of not stamping passports of EU citizens when they cross EU borders.