The megajansky radio burst, FRB 20200428, and other bright radio bursts detected from the Galactic source SGR J1935+2154 suggest that magnetars can make fast radio bursts (FRBs), but the emission site and mechanism of FRB-like bursts are still unidentified. Here, we report the emergence of a radio pulsar phase of the magnetar 5 months after FRB 20200428. Pulses were detected in 16.5 hours over 13 days using the Five-hundred-meter Aperture Spherical radio Telescope, with luminosities of about eight decades fainter than FRB 20200428. The pulses were emitted in a narrow phase window anti-aligned with the x-ray pulsation profile observed using the x-ray telescopes. The bursts, conversely, appear in random phases. This dichotomy suggests that radio pulses originate from a fixed region within the magnetosphere, but bursts occur in random locations and are possibly associated with explosive events in a dynamically evolving magnetosphere. This picture reconciles the lack of periodicity in cosmological repeating FRBs within the magnetar engine model.