The site was organised by Garden Organic's Sowing New Seeds project to conserve and capture the unusual crops found on ethnic community allotments in the Midlands.
The large "organoponico", which took six months of planning, building and gardening, mimics the original urban food gardens of Cuba. It incorporates three global garden beds, a ruined colonial building and a "rum shack" that doubles as an educational space for visitors.
Sowing New Seeds' Anton Rosenfeld said: "Migrant communities have been saving seeds for years that are adapted to UK growing conditions. The people growing them are reaching an age where there is a danger of losing both the seeds and the knowledge. The Exotic Garden is a way of preserving, sharing and educating."
The collection of vegetables in the garden includes Calaloo (amaranth), Chinese stem lettuce, shark fin melon and Dudi.
TV star James Wong, who opened the garden, said: "This project is vital in showing that these lesser-known crops are being grown by ordinary people to eat, and not just by eccentric hobbyists. There must be hundreds more varieties out there and it's important that we learn as much about them as we can."
The Exotic Garden, funded by the National Lottery, Brook Trust, Sheldon Trust and Fund for the Environment & Urban Life, is open to visitors at Ryton Gardens in Warwickshire.