We investigate the scaling properties of a model of passive vector turbulence with pressure in the presence of a large-scale anisotropy. The leading scaling exponents of the structure functions are proven to be anomalous. The anisotropic exponents are organized in hierarchical families growing without bound with the degree of anisotropy. Nonlocality produces poles in the inertial-range dynamics corresponding to the dimensional scaling solution. The increase vs. the Péclet number of the hyperskewness and higher odd-dimensional ratios signals the persistence of anisotropy in the inertial range.