<<Disclaimer: Verify this information before applying it to your situation.>>

Summary re asian fish sauces:  There are indeed brands of fish sauce
which contain wheat protein.  Eeuw!  Why does a company need to
contaminate a perfectly good product made of "aged" fish with icky wheat
proteins??????

Anyway, here are the comments I received:
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3 Crabs (a popular brand) is NOT GF; it has hydrolyzed wheat protein.

We use Squid Brand (also a popular brand) and haven't had any problems
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I eat Thai Kitchen fish sauce and other products.
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Thai Kitchen
"premium extra virgin Fish Sauce" is what the label says -

I'm told some brands are not GF but only picked this up because I was
referred to it as GF.-
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 I use Squid brand which a Vietnamese friend said is the best, most favored
brand.  I do not get a reaction from it and I'm very sensitive, and the
ingredients look fine, so I eat it.  It's up to you, of course; it's not
labeled gf & I didn't check with the manufacturer.
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The fish sauce by KA ME is Gluten Free.   I called the company   LIBERTY RICHTER INC.   and they confirmed it.

Hope you like it.
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