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Even though Long Island followed the fate of the rest of New York when the US won its independence, the island was a loyalist stronghold during the war; what if the British had been able to hold on to it? Some of the people that emigrated to Canada after the ARW could head for Long Island instead, depending on how many people do so several scenarios are possible:
Long Island joins Canada
Long Island becomes a separate subject realm of the Crown
Long Island and the Atlantic provinces of OTL Canada form their own federation