In the Noma guide to fermentation, author Rene Red Redzepi suggests the possibility that south East asain fish sauces may have evolved from the Roman garum saying that there aren’t any mentions of fish sauce before ~600 AD and that it’s possible it was brought to asain by the romans over the Silk Road
Do food historians have any opinions on the origins of fish sauce?
The short answer is that they're probably independent. Or, if they're related, their common ancestor is probably quite old.
While garum and SE Asian fish sauce are the best-known (in the West) fish sauces, they are only two examples of a huge range of ancient and modern fermented fish/seafood products. Even today, there's still a wide range of such things, from Japanese and Korean fish sauces, Korean jeotgal, SE Asian fish sauce, SE Asian prawn/fish pastes (from the Philippines to Burma), Indian, Iranian, and Arabian fish/seafood sauces/pastes, West African fish sauces, and the modern descendant of garum, Italian colatura di alici. Fermented fish sauces/pastes are still used in China, even if soybean paste and soy sauce replaced much use of fermented fish in Chinese cuisine about 2000 years ago.
We don't know the very old history, since we run out of written sources for such sauces. By their first mention in East (China, c. 200BC) and West (5th century BC), they were already well-established. It's possible that there is some very ancient ancestor of such things, but independent invention is quite likely: catch fish, put them in a container, and they will ferment. The trick is to make them ferment into something tasty and safe to eat.
For more detail, see: