The solar wind consists of continuous streams of charged particles that escape into the heliosphere from the Sun, and is split into fast and slow components, with the fast wind emerging from the interiors of coronal holes. Near the ecliptic plane, the fast wind from low-latitude coronal holes is interspersed with a highly structured slow solar wind, the source regions and drivers of which are poorly understood. Here we report extreme-ultraviolet observations that reveal a spatially complex web of magnetized plasma structures that persistently interact and reconnect in the middle corona. Coronagraphic white-light images show concurrent emergence of slow wind streams over these coronal web structures. With advanced global magnetohydrodynamics coronal models, we demonstrate that the observed coronal web is a direct imprint of the magnetic separatrix web (S-web). By revealing a highly dynamic portion of the S-web, our observations open a window into important middle-coronal processes that appear to play a key role in driving the structured slow solar wind.