We study experimentally, in a large-scale basin, the propagation of unidirectional deep water gravity waves stochastically modulated in phase. We observe the emergence of nonlinear localized structures that evolve on a stochastic wave background. Such a coexistence is expected by the integrable turbulence theory for the nonlinear Schr{o}dinger equation (NLSE), and we report the first experimental observation in the context of hydrodynamic waves. We characterize the formation, the properties and the dynamics of these nonlinear coherent structures (solitons and extreme events) within the incoherent wave background. The extreme events result from the strong steepening of wave train fronts, and their emergence occurs after roughly one nonlinear length scale of propagation (estimated from NLSE). Solitons arise when nonlinearity and dispersion are weak, and of the same order of magnitude as expected from NLSE. We characterize the statistical properties of this state. The number of solitons and extreme events is found to increase all along the propagation, the wave-field distribution has a heavy tail, and the surface elevation spectrum is found to scale as a frequency power-law with an exponent --4.5 $pm$ 0.5. Most of these observations are compatible with the integrable turbulence theory for NLSE although some deviations (e.g. power-law spectrum, asymmetrical extreme events) result from effects proper to hydrodynamic waves.
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