Can I store fish sauce at room temperature, and how do I know when it's gone bad?

That advice isn't "wrong" and millions of people keep keep fish sauce in a cabinet for decades. Regarding safety, it's generally OK to store fish sauce at room temperature for years, but that isn't recommended by government worrywarts for best quality. Still Tasty. Pathogens run in fear faced with this stuff, but it can (rarely) develop "offness".

It is conservative to even bring up this article from Slate:

The fish-sauce-making process is carefully controlled to prevent the growth of dangerous bacteria. First, whole, tiny fish (predominantly anchovies) are combined with salt in roughly a three-to-one ratio. The mixture is then tossed into either concrete or wooden tanks. Enzymes contained within the cells of the fish flesh degrade the proteins, fats, and other molecules into amino acids (compounds responsible for the deeply savory flavor in fish sauce), eventually liquifying the solid tissue. The salt accelerates this process, known as protein hydrolysis, by causing the fish cells to open up, which releases the enzymes.

The briny conditions in the tank keep most microbes and all pathogens at bay, but some harmless, extremely salt-tolerant bacteria, known as halophilic bacteria, survive. These occur naturally in the guts, mouth, and skin of the decomposing anchovies. They too break down the fish proteins, yielding various acids, alcohols, and nitrogen compounds that give fish sauce its characteristic cheesy, meaty flavors. But by the end of fish sauce’s six- to twelve-month fermentation period, the salt has managed to kill even the hardiest halophilic bacteria. At this point, the clear, amber liquid in the tanks is filtered, and any sediment is discarded. The fermentation process does such a good job of killing off bacteria that no pasteurization is required.

There is no legal requirement in the U.S. for fish sauce to bear an expiration date, but most manufacturers put dates on the label anyway, since people are more inclined to trust foods that tell us when they should be discarded. These dates tend to be three to four years from the date of manufacture, which is actually on the conservative side in light of industry guidelines. The Handbook of Indigenous Fermented Foods in the ASCA Countries (published by the Association for Science Cooperation in Asia, a science-policy organization) ascribes to the condiment a shelf life of five years.

This is isn’t to say fish sauce will always be good, either before or after the use-by date (though you should always refrigerate it after opening). It can deteriorate in quality over an extended period of time due to chemical reactions, resulting in color changes or the development of “off” flavors. On rare occasions mold or yeast might develop on the inside surface or lip of the bottle where there’s excess moisture and less salt. These growths are usually innocuous, but as with any food, if it looks strange, smells strange, or tastes strange, you should throw it out.

So, grudgingly, I say throw this batch out 'cause it's fuzzy. But don't lose any sleep over it.

Next time buy a bottle that you're likely to use within a couple of years and keep it in the fridge after opening.

In Thailand, they never refrigerate fish sauce--but they go through it so fast, it doesn't have time to go "off". Fish sauce will go off no matter what you do. It gets darker and funkier. If I have a bottle that has gone dark, I throw it away. It's not that it would hurt you to eat it, but for me, the taste is ruined.

I lived in Thailand for nine years and I keep my fish sauce in the refrigerator. It slows down the aging for sure. I don't think it changes the flavor. The slowed, continuous aging sure does--but that happens in and out of the refrigerator alike. If your fish sauce has started to crystallize in the refrigerator, it has been in there too long and should be thrown out anyway.

Also, if I am making a dish that has raw fish sauce (som tum, many of the dipping sauces), I always start with a fresh unopened bottle. For me, even slightly off fish sauce ruins those dishes.