Memes of Production

That's kind of a myth put out to calm the public when H-bombs were new. They have a fusion stage, and fusion doesn't produce fallout, but the fusion isn't solely a source of energy, it's also a source of neutrons to fission an un-enriched uranium tamper that surrounds the warhead, and fissioning uranium produces fallout.

You might be thinking of the Soviet Tsar Bomba test, which was the largest thermonuclear explosion in history, and produced minimal fallout, but wasn't actually the whole bomb as designed. The tamper was replaced with lead for the test (there's no point covering a wide area in fallout if you don't know whether the rest of the bomb works yet), so the output was about half what it would have been otherwise (which balanced out the fact that the bomb was about twice as powerful as expected).

Every country with thermonuclear weapons has a fissionable tamper on them. There's not much point making your nukes half as scary as they could be for the same money. No one's expecting to invade anywhere they've just nuked or to be around to have to deal with the humanitarian crisis afterwards.