  High frequency peaked high redshift blazars, are extreme in the sense that
their spectrum is particularly hard and peaks at TeV energies. Standard
leptonic scenarios require peculiar source parameters and/or a special setup in
order to account for these observations. Electromagnetic cascades seeded by
ultra-high energy cosmic rays (UHECR) in the intergalactic medium have also
been invoked, assuming a very low intergalactic magnetic field (IGMF). Here we
study the synchrotron emission of UHECR secondaries produced in blazars located
in magnetised environments, and show that it can provide an alternative
explanation to these challenged channels, for sources embedded in structured
regions with magnetic field strengths of the order of $10^{-7}$ G. To
demonstrate this, we focus on three extreme blazars: 1ES 0229+200, RGB
J0710+591, and 1ES 1218+304. We model the expected gamma-ray signal from these
sources through a combination of numerical Monte Carlo simulations and solving
the kinetic equations of the particles in our simulations, and explore the
UHECR source and intergalactic medium parameter space to test the robustness of
the emission. We show that the generated synchrotron pair halo/echo flux at the
peak energy is not sensitive to variations in the overall IGMF strength. This
signal is unavoidable in contrast to the inverse Compton pair halo/echo
intensity, which is appealing in view of the large uncertainties on the IGMF in
voids of large scale structure. It is also shown that the variability of blazar
gamma-ray emission can be accommodated by the synchrotron emission of secondary
products of UHE neutral beams if these are emitted by UHECR accelerators inside
magnetised regions.
