Two-stage nuclear weapons get cleaner as they get bigger, because the excess neutrons from the second stage will destroy any long-lived fission products from the first stage. On top of that, all you need is a tamper that doesn't get activated by neutrons and you can push the fallout down to insignificance. This is why "atoms for peace" and such initiatives were not quite as stupid as they sound at the outset, as it would have been possible to build bombs that don't cause much fallout for them.
However, since you can roughly double the power of any two-stage bomb by replacing the tamper with uranium, at minimal expense and with no added mass, but at the cost of producing massive amounts of fallout, real bombs deployed for war generally would have made a lot of fallout.
That's all true of bombs detonated high up in the air where the only elements around are hydrogen, oxygen, nitrogen, carbon, and argon - none of which tend to be neutron activated in a particularly dangerous way. On the ground if you're using bombs for engineering work there are a lot of other elements that do get very upset if you hit them with fusion neutrons.
Even in the context of 3 stage designs there's a huge difference in fallout between airbursts and groundbursts. It's the groundburst fallout that would have killed everyone in Copehagen and Stockholm if WWIII had gone down when the wind was blowing from the east, even with a successful US first strike.
However, since you can roughly double the power of any two-stage bomb by replacing the tamper with uranium, at minimal expense and with no added mass, but at the cost of producing massive amounts of fallout, real bombs deployed for war generally would have made a lot of fallout.