Magnetohydrodynamic turbulence is central to laboratory and astrophysical plasmas, and is invoked for interpreting many observed scalings. Verifying predicted scaling law behavior requires extreme-resolution direct numerical simulations (DNS), with needed computing resources excluding systematic parameter surveys. We here present an analytic generator of realistically looking turbulent magnetic fields, that computes threedimensional (3D) O(1000 3) solenoidal vector fields in minutes to hours on desktop computers. Our model is inspired by recent developments in 3D incompressible fluid turbulence theory, where a Gaussian white noise vector subjected to a nonlinear transformation results in an intermittent, multifractal random field. Our B × C model has only few parameters that have clear geometric interpretations. We directly compare a (costly) DNS with a swiftly B × C-generated realization, in terms of its (1) characteristic sheetlike structures of current density, (2) volume-filling aspects across current intensity, (3) power-spectral behavior, (4) probability distribution functions of increments for magnetic field and current density, structure functions, and spectra of exponents, and (5) partial variance of increments. The model even allows to mimic time-evolving magnetic and current density distributions and can be used for synthetic observations on 3D turbulent data cubes.