> Ketchup was not invented in the United States. It began as a fermented fish sauce — sans tomatoes — in early China. British sailors bought the sauce, called ke-tsiap or ke-tchup by 17th-century Chinese and Indonesian traders, to provide relief from the dry and mundane hardtack and salt pork they ate aboard ship. Over the next couple of centuries, ketchup spread throughout the British Empire, traveling around the world with the navy.
The (surely apocryphal) origin story I heard for Worcestershire sauce was that it was a (failed) attempt to replicate an Indian tamarind chutney, which was abandonded, left to ferment in barrels, and rediscovered.
Supposedly Asian fish sauces are similar to both Worcestershire sauce https://en.wikipedia.org/wiki/Worcestershire_sauce and the ancient Roman garum https://en.wikipedia.org/wiki/Garum . Some say that Colatura di alici https://en.wikipedia.org/wiki/Colatura_di_Alici from Cetara in Italy is the best of them all.