  A detection of bosonic metallic state that breaks the $Z_2$ time-reversal
symmetry has been recently reported in Ba$_{1-x}$K$_x$Fe$_2$As$_2$ with a
doping level $x \approx 0.8$. This is a metallic state of fermionic quadruplets
that breaks time-reversal symmetry. As such, it has no condensate of Cooper
pairs but has a long-range order between fermionic quartets. In the present
manuscript, we investigate the emergence of this phase in a two-component
London model via Monte Carlo simulations as a function of various
intercomponent couplings.
