  Soliton and breather solutions of the nonlinear Schr\"odinger equation (NLSE)
are known to model localized structures in nonlinear dispersive media such as
on the water surface. One of the conditions for an accurate propagation of such
exact solutions is the proper generation of the exact initial phase-shift
profile in the carrier wave, as defined by the NLSE envelope at a specific time
or location. Here, we show experimentally the significance of such initial
exact phase excitation during the hydrodynamic propagation of localized
envelope solitons and breathers, which modulate a plane wave of constant
amplitude (finite background). Using the example of stationary black solitons
in intermediate water depth and pulsating Peregrine breathers in deep-water, we
show how these localized envelopes disintegrate while they evolve over a long
propagation distance when the initial phase shift is zero. By setting the
envelope phases to zero, the dark solitons will disintegrate into two gray-type
solitons and dispersive elements. In the case of the doubly-localized Peregrine
breather the maximal amplification is considerably retarded; however locally,
the shape of the maximal focused wave measured together with the respective
signature phase-shift are almost identical to the exact analytical Peregrine
characterization at its maximal compression location. The experiments,
conducted in two large-scaled shallow-water as well as deep-water wave
facilities, are in very good agreement with NLSE simulations for all cases.
