Measurement of the absolute phase of ultrashort optical pulses in real-time is crucial for various applications, including frequency comb and high-field physics. Modern single-shot techniques, such as dispersive Fourier transform and time-lens, make it possible to investigate non-repetitive spectral dynamics of ultrashort pulses yet do not provide the information on absolute phase. In this work, we demonstrate a novel approach to characterise single-shot pulse-to-pulse stability of the absolute phase with the acquisition rate of 15 MHz. The acquisition rate, limited by the repetition rate of the used free-running mode-locked Erbium-doped fibre laser, substantially exceeds one of the traditional techniques. The method is based on the time-stretch technique. It exploits a simple all-fibre Mach-Zehnder interferometric setup with a remarkable resolution of ∼7.3 mrad. Using the proposed method, we observed phase oscillations in the output pulses governed by fluctuations in the pulse intensity due to Kerr-induced self-phase modulation at frequencies peaked at 4.6 kHz. As a proof-of-concept application of the demonstrated interferometric methodology, we evaluated phase behaviour during vibration exposure on the laser platform. The results propose a new view on the phase measurements that provide a novel avenue for numerous sensing applications with MHz data frequencies.