One of Captain Jack Sparrow's nemeses is the East India Trading Company.
Captain Jack Sparrow is one of the many pirates of the Caribbean. The Caribbean is also called the West Indies.
So what is the East India Trading Company doing out West?
According to a deleted scene referenced on the wiki Will says to Lord Beckett.
Will: [looks away, thinking] What is the East India Trading Company doing in the Caribbean?
Beckett: Well, we are...east of India, [explaining by moving his hand] just...the long way around. [pours a glass full of some sort of wine] After all, when goods don't cross borders, then armies will. [raises glass]
Furthermore, in the Visual Guide (currently sourced from the wiki) it states
It began to expand into the Caribbean when the attention of the company's leaders turned to the area's rich plantations. This expansion was led by its Governor/Chairman Lord Cutler Beckett. The company ultimately expanded so much that any threat posed to it was a threat to Great Britain itself.
This would suggest they were looking to get more than the gold, silk and spices they were currently acquiring in the East Indies.
Unfortunately I've been unable to find copies of the original sources, but I shan't give up.
According to Wikipedia, there were a number of East India Companies:
According to Wikipedia, there were also a number of West India Companies.
So it would be natural to suppose that there could have been a British West India Company that monopolized all trade with British colonies in the Caribbean.
But there didn't seem to be any British West India Company. Did the British East India Company operate in the Caribbean? I don't know.
Fortunately for this question the grand or great union flag flown by American colonists in early 1776 before they declared independence and adopted the stars and stripes was quite similar in design to the British East India Company flag and also influenced the design of the American flag. Thus there has been considerable discussion about the possible influence of the British East India Company flag on the union flag of the colonies.
These discussions point out that the company was legally required to trade between the east and ports in Britain. The tea dumped in the Boston Tea Party had been sold by the Company to dealers on consignment and was shipped to the colonies on ships owned and chartered by other companies and individuals. So this discussion makes it seem unlikely that the British East India Company operated in the West Indies.