  Motivated by the recent observations of small Fermi energies and
comparatively large superconducting gaps, present also on bands not crossing
the Fermi energy (incipient bands) in iron-based superconductors, we analyse
the doping evolution of superconductivity in a four-band model across the
Lifshitz transition including BCS-BEC crossover effects on the shallow bands.
Similar to the BCS case we find that with hole doping the phase difference
between superconducting order parameters of the hole bands changes from $0$ to
$\pi$ through an intermediate $s+is$ state breaking time-reversal symmetry. The
transition however occurs in the region where electron bands are incipient and
chemical potential renormalization in the superconducting state leads to a
significant broadening of the $s+is$ region. We further present the qualitative
features of the $s+is$ state that can be observed in scanning tunnelling
microscopy (STM) experiments also taking incipient bands into account.
