There's even a case that Ketchup is a distant relative, as it started out as South East Asian oyster sauce, was imported to Europe, turned into fermented mushroom sauce, was exported to the colonies, and finally turned into tomato sauce (though originally sometimes with fish in it).
Yes! Search for "mushroom ketchup", and you'll find various examples for sale. Whatever kinds I've had are nice on bread, and really nice with eggs, but I wouldn't want to eat with chips / fries.
You can still sometimes find mushroom ketchup in UK supermarkets. It tastes a bit like Worcester sauce (spicy and 'brown' tasting), but milder as it has no anchovies in it.
