This research presents a coherent structured illumination single-pixel imaging scheme to image objects with complex amplitudes. By utilizing a phase-only spatial light modulator for phase modulation, we can efficiently generate the Hadamard basis structured light and the reference light that interfere with each other to form the coherent structured illumination. Using the 4-step phase-shifting, the spectrum of the object is acquired by detecting the zero-frequency component of the object light with a single-pixel photodetector. The desired complex-amplitude image can be further retrieved by applying an inverse Hadamard transform. The proposed scheme is experimentally demonstrated by imaging two etched glass objects, a dragonfly wing, and a resolution test chart. Benefiting from the phase modulation, this scheme has a high efficiency, a high imaging quality, a high spatial resolution, and a simple and stable configuration to obtain both the phase and amplitude information of the target object. The proposed scheme provides a promising complex-amplitude imaging modality with single-pixel detection. Thus it might find broad applications in optical metrology and biomedical science.