Three examples in the spontaneous symmetry breaking that occurs at a phase transitions:
A ferromagnet below the Curie temperature chooses an axis of quantisation along which all the spins align, giving rise to a macroscopic magnetisation in that direction. Symmetry broken is $SO(3)$ (isotropy of space) to $SO(2)$ (only symmetry about magnetisation axis).
Bose-Einstein condensation: below $T_c$, bosons amass in the same ground state, described by the same wavefunction and with a physical (non-gaugeable) phase. Symmetry broken is $U(1)$.
Higgs mechanism. The complex Higgs doublet chooses a phase, a non-zero vacuum expectation value (VEV) $\propto \mu^2/\nu$. This then determines the mass of the Higgs, the $W^\pm$ and $Z^0$ bosons, and the coupling to fermions. Symmetry broken is $SU(2)_L \times U(1)_Y$ to $U(1)_{em}$
in the absence of any decoherence, coupling to environments and measurements, how is the phase chosen?
Why is not a superposition of all possible ones.
The phase controls the masses of particles, for the higgs. So it's quite important. What caused the field to choose that particular phase during the Higgs phase transition?