Reconstructing audio from a given spectrogram and phase data
A spectrogram is a graphical time-scroll representation of the frequency components in a signal. Sometimes it's called a &quot;waterfall&quot;, because that's what it looks like as it is computed and plotted in real time. The core math behind this is the FFT.
A spectrogram generally shows the strength (or magnitude) of a range of frequency components in a signal (e.g. DC to 20kHz). The FFT has a complex output, one block of values for each block of input. The magnitude of these complex numbers is likely (somehow) stored in the &quot;spectrogram&quot; file, and the phase is stored in the &quot;phases&quot; file.
The trick to reconstructing the original signal from a spectrogram is to inverse the spectrum, and the math behind that is the IFFT.
You will need the magnitude and the phase data from the files to recreate the complex spectrum and then apply it to the IFFT to get the time domain signal back.
You will have to find out

what math was applied to calculate the spectrogram: FFT block size, any log-scale (dB?), any windowing, any sub-sampling etc... and how the data is formatted and organized in the file.
how the phase is calculated and stored (block size, degrees/radians, number format etc...)

As a commenter pointed out, you will have to find out all these details from whoever created the file. But beware that even then it might not be possible to recreate the original signal.
In that case, if you can specify how to generate the spectrogram with different parameters (no windowing, no sub-sampling etc...), and then (re)generate these files, you may stand a chance.
Good luck. Since this is a bit like forensics, you will need it.