  A charge-density wave (CDW) state has a broken symmetry described by a
complex order parameter with an amplitude and a phase. The conventional view,
based on clean, weak-coupling systems, is that a finite amplitude and
long-range phase coherence set in simultaneously at the CDW transition
temperature T$_{cdw}$. Here we investigate, using photoemission, X-ray
scattering and scanning tunneling microscopy, the canonical CDW compound
2H-NbSe$_2$ intercalated with Mn and Co, and show that the conventional view is
untenable. We find that, either at high temperature or at large intercalation,
CDW order becomes short-ranged with a well-defined amplitude that impacts the
electronic dispersion, giving rise to an energy gap. The phase transition at
T$_{cdw}$ marks the onset of long-range order with global phase coherence,
leading to sharp electronic excitations. Our observations emphasize the
importance of phase fluctuations in strongly coupled CDW systems and provide
insights into the significance of phase incoherence in `pseudogap' states.
