The term 'Half a Cocktail' is a combination of two drinks, the Mai TAI and the TAI. The Mai TAI is a rum-based cocktail that originated in California in 1944. It is made with lime juice, orange curaçao, orgeat syrup, and dark rum. The TAI is a tropical drink made with light rum, orange curaçao, orgeat syrup, and lime juice. When combined, these two drinks create a unique flavor that is both sweet and tart.