I think the claim is misleading in several ways. First, it's possible to design a "fusion bomb" that maximizes (neutron) radiation output, leading to substantially higher radiation than from a fission device. "The neutron release generated by a nuclear fusion reaction is intentionally allowed to escape the weapon, rather than being absorbed by its other components" to quote Wikipedia on that. And consequently "It is expected that after neutron weapon explosions the induced radioactivity in the soil surface layer, metal objects or metal constructions is about 10 times higher, than after an atomic [=fission] blast of the same power."
Whether this induced radiation should (not) count as 'fallout', YMMV. As that paper explains, what kind of soil gets radiated matters in such contexts.
But I'm guessing NdGT means to leave those neutron bombs aside from the discussion and focus on "proper hydrogen" weapons that harness the neutrons for the 'biggest boom possible' rather than work through escaping/induced radiation.
'Clean' fusion weapons had been advocated e.g. around 1957.
But I don't think they were actually built like that in significant numbers. Because it's more useful to make the device a 3-state/phase one, where the outer casing is (depleted) Uranium 238. The fusion-generated neutrons are energetic enough to turn this 3rd stage into a fission bomb layer of its own. This e.g. the path that China's "H bomb" programme took (which ironically was easier to find info about on a quick search in English):
The so-called “three-phase” nuclear device at the time was a layer-cake-type device, which consisted of three phases: fission, fusion, and then fission again. During the first phase, the highly-enriched uranium or plutonium core generated the fission explosion. The second phase was the fusion reactions of the layer of solid thermonuclear fuel (lithium deuteride) surrounding the core. Tritium was generated when neutrons from the first phase explosion bombard the lithium. Meanwhile, the high temperature generated by the fission explosion caused the fusion reactions of the deuterium and tritium. The third phase was the fission reaction of the layer of uranium 238 (natural uranium or depleted uranium) surrounding the thermonuclear fuel. The very-high energy neutrons released from the fusion reactions can fission uranium 238. For a layer-cake-type model, additional layers of thermonuclear materials and uranium 238 could be used. [...]
The device 629’s primary was similar to that of the first atomic bomb device 596 tested in 1964, which had a solid uranium 235 core with a uranium deuteride neutron source and no thermonuclear materials. The device 596’s yield was set to be about 20 kilotons, whereas the device 629’s design yield was about 100 kilotons. [Achieved 122kt.] This limited total yield required the secondary to host less thermonuclear materials and use lead metal to replace the natural uranium. [...]
The theoretical design of the device 639 was relatively straightforward, as mainly based on that of the low-yield device 629. The primary would use the same design as the device 629, whereas the secondary only needed a few modifications from the 629, such as adding more lithium deuteride materials to have a full yield and replacing the lead metal with natural uranium. The design yield ranged from one-and-a-half to three megatons. By February, the weapon designers had completed the basic theoretical design of the hydrogen bomb 639 and, by May, the design was finalized and ready to be manufactured.
On June 5, the processing and manufacturing of the first device 639 was complete. Following the two-year plan, Plant 221 in the Qinghai province prepared a total of eight test bombs. [...] Based on post-test analysis of various measurements, including radiochemical analysis of air samples, the explosive power of the hydrogen bomb 639 was estimated at 3.3 megatons.
The only diff here compared to the other source[s] is that China used natural rather than depleted Uranium for the outer casing (of the secondary). This layer is also called a "pusher" on Wikipedia (which also concurs that this was a "three-staged thermonuclear device".)
Alas, I don't have any stats as to how many bombs are like this (3-phase/stage), but it's not at all a given the 'modern' ones will not have this 3rd fission stage, given the research and testing that went into it.
Exactly how much fallout comes out of this 3rd layer stage, I'm not totally sure, but there certainly are publications that claim the exact opposite of what NdGT claims, i.e. that's highly significant:
The fallout produced in a nuclear explosion depends greatly on the type of weapon, its explosive yield, and where it’s exploded. The neutron bomb, although it produces intense direct radiation, is primarily a fusion device and generates only slight fallout from its fission trigger. Small fission weapons like those used at Hiroshima and Nagasaki produce locally significant fallout. But the fission-fusion-fission design used in today’s thermonuclear weapons introduces the new phenomenon of global fallout. Most of this fallout comes from fission of the U-238 jacket that surrounds the fusion fuel. The global effect of these huge weapons comes partly from the sheer quantity of radioactive material and partly from the fact that the radioactive cloud rises well into the stratosphere, where it may take months or even years to reach the ground.
(Italics around 'global fallout' in original.)
N.B. the US army (NUCLEAR MATTERS HANDBOOK 2020 revised) doesn't describe such [designs] as 'three stage'... but it does talk of fission being significant in the secondary as well.
Boosted Weapons[:] A boosted weapon increases the efficiency and yield for a weapon of the same volume and weight when a small amount of fusionable material, such as deuterium or tritium gas, is placed inside the core of a fission device. The immediate fireball, produced by the supercritical mass, has a temperature of tens of millions of degrees and creates enough heat and pressure to cause the nuclei of the light atoms to fuse together. In this environment, a small amount of fusion gas, measured in grams, can produce a huge number of fusion events. Generally, for each fusion event, there is one high-energy neutron produced. These high-energy neutrons then interact with the fissile material, before the weapon breaks apart in the nuclear detonation, to cause additional fission events that would not occur if the fusion gas were not present. This approach to increasing yield is called “boosting” and is used in most modern nuclear weapons to meet yield requirements within size and weight limits. In general, the boosted weapon design is more technically complex than the implosion design and also more efficient.
Staged Weapons[:] A staged weapon [...] normally uses a boosted primary stage and a secondary stage to produce a significantly increased yield. In the first stage, a boosted fission device releases the energy of a boosted weapon, which includes a large number of X-rays. The X-rays transfer energy to the secondary stage, causing fusionable material in the secondary to undergo fusion, which releases large numbers of high-energy neutrons. These neutrons, in turn, interact with fissionable material in the secondary to cause a huge number of fission events, thereby significantly increasing the yield of the whole weapon. The two-stage weapon design is more technically complex than any other weapon design. For a given size, it can produce a much larger yield than any other design.
There's no mention there of the 'clean' designs that would not use fission in the secondary.
They do discuss fallout quite bit though. I'm not sure what would be the most relevant part to quote. Maybe this, which partly agrees with the MIT quote I emphasized previously:
If the detonation is a true air burst in which the fireball does not interact with the ground or any significant structure, the size and heat of the fireball causes it to retain almost all of the weapon debris, usually one or at most a few tons of material, as it moves upward in altitude and downwind. In this case, very few particles fall to the ground at any moment and no significant radioactive hot-spot on the ground is caused by the fallout. The fireball rises to become a long-term radioactive cloud. The cloud travels with the upper atmospheric winds and circles the hemisphere several times, over a period of months, before it dissipates completely. Most of the radioactive particles decay to stable isotopes before falling to the ground. The particles that reach the ground are distributed around the hemisphere at the latitudes of the cloud travel route. Even though there would be no location receiving a hazardous amount of fallout radiation, certain locations on the other side of the hemisphere could receive more fallout, which is measurable with radiation detectors, than the area near the detonation. This phenomenon is called worldwide fallout.
(Italics emphasis in original again.) So, I guess it's a matter of perspective whether you care about local or global fallout. However, NdGT doesn't seem to have this tradeoff in mind, at least based on the short snippet quoted by the OP. The Handbook then discusses surface detonations:
As large and hot as the fireball is (1-kt detonation produces a fireball almost 200 feet in diameter and tens of millions of degrees), it has no potential to carry thousands of tons of material. Thus, as the fireball rises, it begins to release a significant amount of radioactive dust, which falls to the ground and produces a radioactive fallout pattern around GZ and in areas downwind. The intensity of radioactivity in this fallout area would be hazardous for weeks. This is called early fallout, caused primarily by a surface-burst detonation regardless of the weapon design. Early fallout would be a concern in the case of employment of a nuclear threat device during a terrorist attack. [...] If a detonation is a surface or near-surface burst, early fallout would be a significant radiation hazard around GZ and downwind.
(Italics of "early fallout" in original, bold around "regardless of the weapon design" mine.) So, I guess the issues that really matter wrt fallout are somewhat different than what NdGT envisaged.
