  Superconductivity is described by an order parameter of second order (i.e.\
quadratic) in fermionic fields, representing electrons forming pairs. However,
in systems with electronic pairing instabilities in several bands there might
exist a phase that is different by symmetry from both normal metal and
superconductor. This non-superconducting state is described by an order
parameter that is of fourth order in fermionic fields. Here, we report the
observation of such a phase in the superconductor Ba$_{1-x}$K$_x$Fe$_2$As$_2$.
This novel state can be characterized as a bosonic $Z_2$ metal, which
spontaneously breaks time-reversal symmetry as a consequence of the formation
of non-condensed Cooper-pairs. In this state, the phases of the gaps in
different bands are disordered and superconductivity is absent. However, the
gaps do not fluctuate independently and there is a broken time-reversal
symmetry due to a non-trivial difference between the fluctuating phases in
different bands. Hence, this state can be described as an ordering which is
fourth order in fermions, i.e.\ consists of correlations between pairs of
noncondensed Cooper pairs of fermions in different bands. When temperature is
decreased, the system undergoes a superconducting phase transition and enters a
$\mathrm{U}(1)\times Z_2$ superconducting state. This is in precise agreement
with earlier muon spin rotation experiments on the formation of $s+\mathrm{i}s$
or $s+\mathrm{i}d$ superconducting states at low temperatures in
Ba$_{1-x}$K$_x$Fe$_2$As$_2$ for the present doping. We demonstrate that a
bosonic $Z_2$ metal shows a number of novel physical properties.
