There are only two widely used "New World" spices, Allspice and Vanilla.
Allspice are the berries of a tree found in tropical America primarily Jamaica.
http://www.pbase.com/selvin/image/27616320
Vanilla is derived from the dried fermented bean pod of an orchid plant. It is the only edible fruit of an orchid. Some orchid flowers are edible.
For many years Colonial Imperialist Nations (read British, French, and Dutch) unsuccessfully tried to grow vanilla beans in their old world colonies.
The pollinating insect (Mexican Bee [Melipone]) and hummingbirds were not present in the old world plantations so the vanilla pods could not be created.
Efficient hand pollination was finally perfected by a former slave (Edmund Albius) on the French Island of Reunion in 1841.
Vanilla beans were successfully raised breaking the new world monopoly held primarily by Mexico.