"Growing allspice bushes is easy. They adapt well to most soils and thrive in a variety of climates. The shrubs are hardy in U.S. Department of Agriculture hardiness zones 5b through 10a."
That is a quote from Gardening Know How. That is for the Carolina allspice bush.
The Allspice pimenta is the spice we are looking for, which is topical. We would need a hot house, or grow it indoors. I'm sure that someone would be able to grow the male and female Allspice in a protected area to be able to harvest the berries for the spice.
Otherwise, you can grow the former and crush the dried leaves for aromatic use.
