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Shocks form the basis of our understanding for the density and velocitystatistics of supersonic turbulent flows, such as those found in the coolinterstellar medium (ISM). The variance of the density field,$\sigma^2_{\rho/\rho_0}$, is of particular interest for molecular clouds (MCs),the birthplaces of stars in the Universe. The density variance may be used toinfer underlying physical processes in an MC, and parameterises the starformation (SF) rate of a cloud. However, models for $\sigma^2_{\rho/\rho_0}$all share a common feature -- the variance is assumed to be isotropic. Thisassumption does not hold when a trans/sub-Alfv\'enic mean magnetic field,$\vec{B}_0$, is present in the cloud, which observations suggest is relevantfor some MCs. We develop an anisotropic model for $\sigma_{\rho/\rho_0}^2$,using contributions from hydrodynamical and fast magnetosonic shocks thatpropagate orthogonal to each other. Our model predicts an upper bound for$\sigma_{\rho/\rho_0}^2$ in the high Mach number $(\mathcal{M})$ limit assmall-scale density fluctuations become suppressed by the strong $\vec{B}_0$.The model reduces to the isotropic $\sigma_{\rho/\rho_0}^2-\mathcal{M}$relation in the hydrodynamical limit. To validate our model, we calculate$\sigma_{\rho/\rho_0}^2$ from 12~high-resolution, three-dimensional,supersonic, sub-Alfv\'enic magnetohydrodynamical (MHD) turbulence simulationsand find good agreement with our theory. We discuss how the two MHD shocks maybe the bimodally oriented over-densities observed in some MCs and theimplications for SF theory in the presence of a sub-Alfv\'enic $\vec{B}_0$. Bycreating an anisotropic, supersonic density fluctuation model, this study pavesthe way for SF theory in the highly anisotropic regime of interstellarturbulence.

A multi-shock model for the density variance of anisotropic, highly-magnetised, supersonic turbulence