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In 1816, the United Kingdom annexed the islands, ruling them from the Cape Colony in South Africa. This is reported to have primarily been a measure to ensure that the French would be unable to use the islands as a base for a rescue operation to free Napoleon Bonaparte from his prison on Saint Helena. The occupation also prevented the United States from using Tristan da Cunha as a cruiser base, as it had during the War of 1812.

where did the UK rule the islands from?
South Africa