We advance a protocol to measure the complex spatial degree of coherence of a partially coherent light field obeying Gaussian statistics through a generalized Hanbury Brown--Twiss (HBT) experiment. The proposed generalized HBT experiment amounts to combining a partially coherent field with a pair of coherent reference fields and measuring the intensity-intensity cross-correlation of the cumulative field. The real or imaginary part of the complex spatial degree of coherence can be extracted directly from the intensity-intensity cross-correlation by adjusting the phase delay between the two reference fields to be $0$ or $\ensuremath{\pi}/2$. We test our method by carrying out a proof-of-principle experiment to measure the complex spatial degree of coherence of inhomogeneous light fields obeying Gaussian statistics. We find excellent agreement between the experimental results and our general theory. We show that our approach is extremely robust against the environmental fluctuations (e.g., the atmospheric turbulence) during the measurement. We demonstrate experimentally that a moving target hiding behind a ground-glass disk can be tracked with the aid of our complex-spatial-coherence measurement protocol.