(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 systems 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.
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