  We report characteristic vortex configurations in $s+id$ superconductors with
time reversal symmetry breaking, exposed to magnetic field. A vortex in the
$s+id$ state tends to have an opposite phase winding between $s-$ and $d-$wave
condensates. We find that this peculiar feature together with the competition
between $s-$ and $d-$wave symmetry results in three distinct classes of
vortical configurations. When either $s-$ or $d-$ condensate absolutely
dominates, vortices form a conventional lattice. However, when one condensate
is relatively dominant, vortices organize in chains that exhibit skyrmionic
character, separating the chiral components of the $s \pm id$ order parameter
into domains within and outside the chain. Such skyrmionic chains are found
stable even at high magnetic field. When $s-$ and $d-$ condensates have a
comparable strength, vortices split cores in two chiral components to form
full-fledged skyrmions, i.e. coreless topological structures with an integer
topological charge, organized in a lattice. We provide characteristic magnetic
field distributions of all states, enabling their identification in e.g.
scanning Hall probe and scanning SQUID experiments. These unique vortex states
are relevant for high-T$_c$ cuprate and iron-based superconductors, where the
relative strength of competing pairing symmetries is expected to be tuned by
temperature and/or doping level, and can help distinguish $s+is$ and $s+id$
superconducting phases.
