Nondestructive microscale investigation of objects is an invaluable tool in life and materials sciences. Currently, such investigation is mainly performed with X-ray laboratory systems, which are based on absorption-contrast imaging and cannot access the information carried by the phase of the X-ray waves. The phase signal is, nevertheless, of great value in X-ray imaging as it is complementary to the absorption information and in general more sensitive to visualize features with small density differences. Synchrotron facilities, which deliver a beam of high brilliance and high coherence, provide the ideal condition to develop such advanced phase-sensitive methods, but their access is limited. Here we show how a small modification of a laboratory setup yields simultaneously quantitative and 3D absorption and phase images of the object. This single-shot method is based on correlation of X-ray near-field speckles and represents a significant broadening of the capabilities of laboratory-based X-ray tomography.