In a fission bomb, the fallout consists of fission-decay fragments, which are nuclei that can have long enough half-lives to be transported by winds. Fusion bombs are basically the same idea, because they use fission triggers.
and is ionizing radiation capable of radiating materials for a long time ?
In theory, yes, e.g., exposure to neutrons in reactors can be used to intentionally produce radioactive isotopes. In practice, although nuclear bombs must produce this kind of artificial transmuation of the surrounding matter (e.g., they do emit neutrons), I think there isn't enough of this kind of process to contribute noticeably to the fallout.
Matter-antimatter annihilation from a hypothetical macroscopic explosion would produce the same particles as proton-antiproton annihilation in microscopic quantities in accelerator experiments. You get high-energy (~100 MeV) gammas, medium-energy (e.g., 511 keV) gammas, pions, muons, and neutrinos. The neutrinos fly off harmlessly and undetectably into outer space. Matter is nearly transparent to the high-energy gammas; the downward-emitted ones are absorbed somewhere underground. The medium-energy gammas are absorbed in nearby matter. The pions and muons are unstable and decay quickly into stable particles such as electrons. Nothing long-lived is produced.