Black garlic is not a variety of garlic. It is a product made from garlic.
From Wikipedia:
Black garlic is a type of caramelized garlic (a Maillard reaction, not fermentation) first used as a food ingredient in Asian cuisine. It is made by heating whole bulbs of garlic (Allium sativum) over the course of several weeks, a process that results in black cloves. The taste is sweet and syrupy with hints of balsamic vinegar or tamarind. Black garlic's popularity has spread to the United States as it has become a sought-after ingredient used in high-end cuisine.
EDITED:
Here is a link on how to make your own:
https://garlicshaker.com/garli...
