Why would stft be expected to be restricted to real values?
Stft is just a discrete fourier transform at each step. The result of DFT on a real valued signal is a set of complex coefficients that are skew conjugate symmetric: N real inputs representable in N/2 complex coefficients. That preserves information capacity. In order for the DFT results to be real valued as you seem to think they should be, then the original N points of information would have to be half redundant.
The complex portion of the DFT coefficients encode phase information. Real-life signals almost have some phase information.
