We study the ionization and kinematics of the ionized gas in the nuclear region of the barred Seyfert 2 galaxy NGC 5643 using MUSE integral field observations in the framework of the Measuring Active Galactic Nuclei Under MUSE Microscope (MAGNUM) survey. The data were used to identify regions with different ionization conditions and to map the gas density and the dust extinction. We find evidence for a double-sided ionization cone, possibly collimated by a dusty structure surrounding the nucleus. At the center of the ionization cone, outflowing ionized gas is revealed as a blueshifted, asymmetric wing of the OIII emission line, up to projected velocity v10 ~ −450 km s-1. The outflow is also seen as a diffuse, low-luminosity radio and X-ray jet, with similar extension. The outflowing material points in the direction of two clumps characterized by prominent line emission with spectra typical of HII regions, located at the edge of the dust lane of the bar. We propose that the star formation in the clumps is due to positive feedback induced by gas compression by the nuclear outflow, providing the first candidate for outflow-induced star formation in a Seyfert-like, radio-quiet AGN. This suggests that positive feedback may be a relevant mechanism in shaping the black hole-host galaxy coevolution.