Realizing high resolution across large volumes is challenging for 3D imaging techniques with high-speed acquisition. Here, we describe a new method for 3D intensity and phase recovery from 4D light field measurements, achieving enhanced resolution via Fourier ptychography. Starting from geometric optics light field refocusing, we incorporate phase retrieval and correct diffraction artifacts. Further, we incorporate dark-field images to achieve lateral resolution beyond the diffraction limit of the objective (5× larger NA) and axial resolution better than the depth of field, using a low-magnification objective with a large field of view. Our iterative reconstruction algorithm uses a multislice coherent model to estimate the 3D complex transmittance function of the sample at multiple depths, without any weak or single-scattering approximations. Data are captured by an LED array microscope with computational illumination, which enables rapid scanning of angles for fast acquisition. We demonstrate the method with thick biological samples in a modified commercial microscope, indicating the technique’s versatility for a wide range of applications.