Streamers and pseudostreamers structure the corona at the largest scales, as seen in both eclipse and coronagraph white-light images. Their inverted-goblet appearance encloses broad coronal loops at the Sun and tapers to a narrow radial stalk away from the star. The streamer associated with the global solar dipole magnetic field is long-lived, predominantly contains a single arcade of nested loops within it, and separates opposite-polarity interplanetary magnetic fields with the heliospheric current sheet anchored at its apex. Pseudostreamers, on the other hand, are transient, enclose double arcades of nested loops, and separate like-polarity fields with a dense plasma sheet. We use numerical magnetohydrodynamic simulations to calculate, for the first time, the formation of pseudostreamers in response to photospheric magnetic-field evolution. Convective transport of a minority-polarity flux concentration, initially positioned under one side of a streamer, through the streamer boundary into the adjacent, pre-existing coronal hole forms the pseudostreamer. Interchange magnetic reconnection at the overlying coronal null point(s) governs the development of the pseudostreamer above – and of a new, satellite coronal hole behind – the moving minority polarity. The reconnection dynamics liberate coronal-loop plasma that can escape into the heliosphere along so-called separatrix-web (“S-Web”) arcs, which reach far from the heliospheric current sheet and the solar equatorial plane, and can explain the origin of high-latitude slow solar wind. We describe the implications of our results for in-situ and remote-sensing observations of the corona and heliosphere as obtained, most recently, by Parker Solar Probe and Solar Orbiter.