  The London model describes strongly type-2 superconductors as massive vector
field theories, where the magnetic field decays exponentially at the length
scale of the London penetration length. This also holds for isotropic
multi-band extensions, where the presence of multiple bands merely renormalises
the London penetration length. We show that, by contrast, the magnetic
properties of anisotropic multi-band London models are not this simple, and the
anisotropy leads to the inter-band phase differences becoming coupled to the
magnetic field. This results in the magnetic field in such systems having N+1
penetration lengths, where N is the number of field components or bands. That
is, in a given direction, the magnetic field decay is described by N+1 modes
with different amplitudes and different decay length scales. For certain
anisotropies we obtain magnetic modes with complex masses. That means that
magnetic field decay is not described by a monotonic exponential increment set
by a real penetration length but instead is oscillating. Some of the
penetration lengths are shown to diverge away from the superconducting phase
transition when the mass of the phase-difference mode vanishes. Finally the
anisotropy-driven hybridization of the London mode with the Leggett modes can
provide an effectively non-local magnetic response in the nominally local
London model. Focusing on the two-component model, we discuss the magnetic
field inversion that results from the effective non-locality, both near the
surface of the superconductor and around vortices. In the regime where the
magnetic field decay becomes non-monotonic, the multiband London superconductor
is shown to form weakly-bound states of vortices.
