  We apply the static auxiliary field Monte Carlo approach to study phase
correlations of the pairing fields in a microscopic model with spin-singlet
pairing interaction. We find that the short- and long-range phase correlations
are well captured by the phase mutual information, which allows us to construct
a theoretical phase diagram containing the uniform $d$-wave superconducting
region, the phase fluctuating region, the local pairing region, and the
disordered region. We show that the gradual development of phase coherence has
a number of consequences on spectroscopic measurements, such as the development
of the Fermi arc and the anisotropy in the angle-resolved spectra, scattering
rate, entropy, specific heat, and quasiparticle dispersion, in good agreement
with experimental observations. For strong coupling, our Monte Carlo simulation
reveals an unexpected charge-4$e$ plaquette state with $d$-wave bonds, which
competes with the uniform $d$-wave superconductivity and exhibits a U-shaped
density of states.
