The joke's premise is essentially: "the rest of the world is so ignorant about the Caribbean (i.e. the only thing they know about it is Rihanna), we can tell them Rihanna is ours and they'll never know the difference.
Since that Tweet yesterday, Jamaican Twitter managed to get the hashtag #RihannaIsJamaican trending. Participants have constructed Jamaican Rihanna's entire biography, edited her Wikipedia page, and are sharing photoshopped versions of her passport and Jamaican currency. Essentially, Jamaica is gaslighting Barbados.
Barbadian Twitter, however, will under no circumstances will allow for their queen's heist. They've launched a counter-attack of their own and are even attempting burgle Jamaican dancehall legend Sean Paul.
Rihanna is from BARBADOS🇧🇧 not Jamaica🇯🇲! Who started this #RihannaIsJamaican shit I wanna fight https://t.co/UdQ8xBJU7P