  Detailed knowledge of the phase diagram and the nature of the competing
magnetic and superconducting phases is imperative for an understanding of the
physics of iron-based superconductivity. Here, we show using thermodynamic
probes that the phase diagram of the first discovered, and highest Tc, 122-type
material, Ba1-xKxFe2As2 is in fact much richer than previously reported. Inside
the usual stripe-type magnetic order with C2-symmetry, we find a small pocket
of a tetragonal, C4-symmetric phase, which surprisingly reverts back to the
C2-phase at or slightly below the superconducting transition. This re-entrance
to a low-temperature orthorhombic state induced by superconductivity is
discussed in terms of competition of the two magnetic phases with
superconductivity and is illustrated by the measured changes in the electronic
entropy of the system. Using our thermodynamic data, we make predictions about
how the phase diagram of these competing orders will change under pressure.
