  (Abridged) Context. Most massive stars are located in multiple stellar
systems. Magnetic fields are believed to be essential in the accretion and
ejection processes around single massive protostars. Aims. Our aim is to unveil
the influence of magnetic fields in the formation of multiple massive stars, in
particular on the fragmentation modes and properties of the multiple
protostellar system. Methods. Using RAMSES, we follow the collapse of a massive
pre-stellar core with (non-ideal) radiation-(magneto-)hydrodynamics. We choose
a setup which promotes multiple stellar system formation. Results. In the
purely hydrodynamical models, we always obtain (at least) binary systems. When
more than two stars are present, their gravitational interaction triggers
mergers until there are two stars left. The following gas accretion increases
their orbital separation and hierarchical fragmentation occurs so that both
stars host a comparable disk and stellar system which then form similar disks
as well. We identify several modes of fragmentation: Toomre-unstable disk
fragmentation, arm-arm collision and arm-filament collision. Disks grow in size
until they fragment and become truncated as the newly-formed companion gains
mass. When including magnetic fields, the picture evolves: the primary disk
produces less fragments, arm-filament collision is absent. Magnetic fields
reduce the initial orbital separation but do not affect its further evolution,
which is mainly driven by gas accretion. With magnetic fields, the growth of
individual disks is regulated even in the absence of fragmentation or
truncation. Conclusions. Hierarchical fragmentation is seen in unmagnetized and
magnetized models. Magnetic fields, including non-ideal effects, are important
because they remove certain fragmentation modes and limit the growth of disks,
which is otherwise only limited through fragmentation.
