Is there any way of retrieving phase information with only access to real part of successive FFT frames?
webaudio isn't returning the real part of the complex FFT transform, but just the log magnitude of the FFT result.  From just this result alone, of the fluctuations between the results of multiple frames, you can't retrieve the phase information.

However, if you can get at the input data and decompose this input vector into a symmetric vector plus an anti-symmetric vector, the magnitude of the FFT of the symmetric vector will be equivalent to scaled real components of the FFT of the original input vector, and the magnitude of the FFT of the anti-symmetric vector will be equal scaled imaginary components of the FFT of the original input vector.  Thus, the decomposition plus 2 magnitude-only FFTs will allow you to determine both the magnitude and phase of the FFT components.

v_even = v + v_reversed
v_odd  = v - v_reversed