What is the origin of fish sauce in asia?

It was called garum, and indeed the ancient Romans used it, as did the ancient Greeks:

Garum was prepared from the intestines of small fishes through the process of bacterial fermentation. Fishermen would lay out their catch according to the type and part of the fish, allowing makers to pick the exact ingredients they wanted. The fish parts were then macerated in salt, and cured in the sun for one to three months. The mixture fermented and liquified in the dry warmth, with the salt inhibiting the common agents of decay. Garum was the clear liquid that formed on the top, drawn off by means of a fine strainer inserted into the fermenting vessel. The sediment or sludge that remained was allec. Concentrated decoctions of aromatic herbs might be added. Flavors would vary according to the locale, with ingredients sometimes from in-house gardens.

Like the fish sauce of today, it was extremely high in glutamic acid, AKA umami.

In Eurasian Sensation the author says:

There appears to be no historical mention of fish sauce being used in Asia before the Early Middle Ages in Europe, which oddly enough is around the same time its use was dwindling in the remnants of the Roman Empire.

That begs a question, right? Did the concept travel, or did Asians develop it independently? Greater minds than mine are working to definitively answer that question:

...flowed from west to east and was eagerly adopted by Asians on the Silk Road. The recipes for garum changed and adapted as they moved east and became nuoc mam and nam pla according to cultural preferences and what gifts the Asian seas had to offer. Archaeologists and food scientists are working to confirm these flows and linkages...

The production of garum and Asian fish sauce is virtually the same as well. Fresh fish and salt in some proportion (recipes vary widely from 5:1 to 2:1) are layered in barrels, clay pits or earthenware crocks. Because of their large size, tuna were cut up before fermenting to prevent putrefaction, but most fish – especially the small species are processed whole and intact. In Rome, oregano and other herbs were added at the production phase for both flavor and to suppress bacterial overgrowth, but this step is largely omitted in the production of Asian sauces. The vats are then left to ferment – sometimes covered or sometimes uncovered (Carthaginian and Roman) in the heat and stirred every few days to a week to ensure even enzymatic digestion of the fish.

From The Silk Road Gourmet