The phase dynamics of a semiconductor laser with optical feedback is studied by construction of the Hilbert phase from its experimentally measured intensity time series. The Hurst exponent is evaluated for the phase fluctuations and grows from 0.5 to approximately 0.7 (indicating fractional Brownian motion) as the feedback strength is increased. A comparison with numerical computations based on a delay-differential equation model shows excellent agreement and reveals the relative roles of spontaneous emission noise and deterministic dynamics for different feedback strengths.