  We introduce a system composed of two (2+1)-dimensional baby-skyrmion models
(BSMs) set on parallel planes and linearly coupled by tunneling of fields. This
system can be realized in a dual-layer ferromagnetic medium. Unlike dual-core
models previously studied\ in nonlinear optics and BEC, here the
symmetry-breaking bifurcation (SBB) in solitons (baby skyrmions) occurs with
the increase of the inter-core coupling ($\kappa$), rather than with its
decrease, due to the fact that even in the uncoupled system neither core may be
empty. Prior to the onset of the symmetry breaking between the two components
of the solitons, they gradually separate in the lateral direction, due to the
increase of $\kappa$, which is explained in an analytical form by means of an
effective interaction potential. Such evolution scenario are produced for the
originally symmetric states with topological charges in the two cores
$\left(B^{(1)},B^{(2)}\right)$, with $% B^{(1)}=B^{(2)}=1,2,3,4$. The evolution
of mixed states, of the $\left(1,2\right)$ and $\left(2,4\right)$ types, with
the variation of $\kappa$ is studied too.
