A well working fission bomb already have very little fallout. If the device works as designed then all fission material gets converted to stable isotopes. It is when the detonation does not go as planned that you get large amounts of half-fissioned material that forms the radioactive fallout. This is generally why some test sites became contaminated while others were not.
I am not quite sure what effects a fusion reaction have on this though. I would presume that the fusion detonation could eliminate some of the fallout from the fission reaction by bombarding it with neutrons. It is also not quite right that fusion reactions have no fallout. Similar to a fission bomb in a perfect scenario all the material end up as stable helium but where things are not perfect you can get radioactive tritium or isotopes of lithium and other elements. And the neutrons can create a lot of radioactive elements in the material around the bomb.
In any way the amount of fallout from a hydrogen bomb compared to its yield is a lot lower, almost negligible. But it does suffer from the same issue as atom bombs in that failed tests can still have considerable fallout.
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