Abstract The majority of stars are in binary/multiple systems. How such systems form in turbulent, magnetized cores of molecular clouds in the presence of non-ideal MHD effects remains relatively under-explored. Through ATHENA++-based non-ideal MHD AMR simulations with ambipolar diffusion, we show that the collapsing protostellar envelope is dominated by dense gravo-magneto-sheetlets, a turbulence-warped version of the classic pseudodisk produced by anisotropic magnetic resistance to the gravitational collapse, in agreement with previous simulations of turbulent, magnetized single-star formation. The sheetlets feed mass, magnetic fields, and angular momentum to a Dense ROtation-Dominated (DROD) structure, which fragments into binary/multiple systems. This DROD fragmentation scenario is a more dynamic variant of the traditional disk fragmentation scenario for binary/multiple formation, with dense spiral filaments created by inhomogeneous feeding from the highly structured larger-scale sheetlets rather than the need for angular momentum transport, which is dominated by magnetic braking. Provided that the local material is sufficiently demagnetized, with a plasma-β of 10 or more, collisions between the dense spiraling filaments play a key role in facilitating gravitational collapse and stellar companion formation by pushing the local magnetic Toomre parameter Qm below unity. This mechanism can naturally produce in situ misaligned systems on the 100-au scale, often detected with high-resolution ALMA observations. Our simulations also highlight the importance of non-ideal MHD effects, which affect whether fragmentation occurs and, if so, the masses and orbital parameters of the stellar companions formed.