Magnetohydrodynamic turbulence is central to laboratory and astrophysical
plasmas, and is invoked for interpreting many observed scalings. Verifying
predicted scaling law behaviour 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 3D $\mathcal{O}({1000}^{3})$
solenoidal vector fields in minutes to hours on desktops. Our model is inspired
by recent developments in 3D incompressible fluid turbulence theory, where a
Gaussian white noise vector subjected to a non-linear transformation results in
an intermittent, multifractal random field. Our $B\times C$ model has only few
parameters that have clear geometric interpretations. We directly compare a
(costly) DNS with a swiftly $B\times C$ -generated realization, in terms of its
(i) characteristic sheet-like structures of current density, (ii)
volume-filling aspects across current intensity, (iii) power-spectral
behaviour, (iv) probability distribution functions of increments for magnetic
field and current density, structure functions, spectra of exponents, and (v)
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.