What are the sources of antimatter particles in the magnetosphere?
Matter and antimatter particles are always produced as a pair. This particle physics' process is called pair production. If antimatter is detectable, it means that the
antiparticle of a matter-antimatter-particle-pair was able to survive long enough to be detected, for instance due to relativistic effects, as explained e.g. by the Hyperphysics page on muons:

The measurement of the flux of muons at the Earth's surface produced an early dilemma because many more are detected than would be expected, based on their short half-life of 1.56 microseconds. This is a good example of the application of relativistic time dilation to explain the increased particle range for high-speed particles.

You will now correctly say that a muon is not antimatter, but I just used the muon as example that some particles (or antiparticles) may be detected even if they have a short half-life.
So, what are the sources of antimatter in the magnetosphere? Strictly speaking, the source (in the sense of &quot;location&quot;) of pair creation will probably be in the magnetosphere itself, as said by atmosphericprisonescape:

Anywhere where highly energetic reactions take place whose center-of-mass energy surpasses the rest-mass energy of 2*times some particle, this particle and its anti-particle counterpart can be created.

I suppose the real question is: How is their enough energy for frequent pair production in the magnetosphere?