Cayman Islands is island group in Caribbean Sea, nearly one-half of the way from Cuba to Honduras. Here aren't distilleries; rum is blended from imported one. Here is list of companies whose rum bottle labels I have in my collection:

Blackbeard's Liquor / Grand Cayman Bottlers & Blenders(10)
Island Companies LTD(1)
Rum Company (West Indies) Ltd., George Town(2)
Seven Fathoms Distillery(0)
Tortuga Rum Company Ltd.(16)