The short answer: Probably independent.
Garum and SE Asian fish sauce are the best-known (in the West) of a wide range of fish sauces and fish pastes and other fermented seafood products used for flavour. In East Asia, fermented fish sauces and pastes were the standard sauces before the development of soy sauce and soybean paste which largely displaced them (fish sauces are still used in Japan and Korea, and also a broad range of fermented seafoods). (Fermented meat sauces were also common, and also largely replaced by soy). As well as East Asia, SE Asia, and the Mediterranean, fish sauces/pastes were, and still are, widespread along the Indian Ocean coast (including East Africa) and in Atlantic Africa. Such sauces/pastes are not confined to coastal areas, either - river fish can also be used, and fish sauces are found through Mesopotamia, in the Himalayas, etc.
In both East and West, the fish sauce tradition is over 2,000 years old. The oldest Chinese texts mentioning it date to about 200BC, but fish sauce probably has a much longer history in China. In the Mediterranean, fish sauce appears in literature in the 5th century (not for its use in cooking per se, but rather for its famed aroma (or stench, some would say)). There isn't any great mystery in fermented fish products being independently discovered in multiple places, since fish ferments rather readily. Mediterranean, E/SE Asian, South Asia, and West African fish sauces/pastes are probably independent developments.
Fish sauce didn't quite disappear from the Mediterranean. A modern descendant of garum still exists: colatura di alici is still made in Italy, with a continuous tradition going back to Antiquity.
For more detail, see:
/u/wotan_weevil had an excellent answer to this same question. It's quite in depth.