You do not have to visit Antigua & Barbuda while making a citizenship-by-investment application. If you are interested in becoming a resident as well as a citizen, once you become a citizen, you can reside in Antigua whenever you wish. If you spend a minimum of six months per year there, you will be considered a permanent resident and taxpayer. I would advise consulting with a licensed agent for the citizenship program as they will be best placed to advise you on all the residency and citizenship options and benefits.
