  (Abridged) Context. Massive stars form in magnetized and turbulent
environments, and are often located in stellar clusters. Their accretion
mechanism, as well as the origin of their system's stellar multiplicity are
poorly understood. Aims. We study the influence of both magnetic fields and
turbulence on the accretion mechanism of massive protostars and their
multiplicity. Methods. We present a series of four Radiation-MHD simulations of
the collapse of a massive magnetized, turbulent core of 100 $M_\odot$ with the
AMR code Ramses, including a hybrid radiative transfer method for stellar
irradiation and ambipolar diffusion. We vary the Mach and Alfvenic Mach numbers
to probe sub- and superalfvenic turbulence as well as sub- and supersonic
turbulence regimes. Results. Subalfvenic turbulence leads to single stellar
systems while superalfvenic turbulence leads to binary formation from disk
fragmentation following spiral arm collision, with mass ratios of 1.1-1.6. In
those runs, infalling gas reaches the individual disks via a transient
circumbinary structure. Magnetically-regulated, thermally-dominated (plasma
beta $\beta>1$), Keplerian disks form in all runs, with sizes 100-200 AU and
masses 1-8 $M_\odot$. The disks around primary and secondary sink particles
share similar properties. We observe higher accretion rates onto the secondary
stars than onto their primary star companion. The primary disk orientation is
found to be set by the initial angular momentum carried by turbulence.
Conclusions. Small (300 AU) massive protostellar disks as those frequently
observed nowadays can only be reproduced so far in the presence of (moderate)
magnetic fields with ambipolar diffusion, even in a turbulent medium. The
interplay between magnetic fields and turbulence sets the multiplicity of
stellar clusters. A plasma beta $\beta>1$ is a good indicator of streamers and
disks.
