We present the first microwave photonic phase shifter using stimulated Brillouin scattering (SBS) on-chip. The unique ability of SBS to generate both narrowband gain and loss resonances allows us to achieve low ±1.5 dB amplitude fluctuations, which is a record for integrated devices, along with 240° continuously tunable phase shift. Contrary to previous SBS-based approaches, the phase shift tuning mechanism relies on tuning the power, not the frequency, of two SBS pumps, making it more suited to on-chip implementations. We finally demonstrate that SBS pump depletion leads to amplitude response fluctuations, as well as increasing the insertion loss of the phase shifter. Advantageously, shorter integrated platforms possess higher pump depletion thresholds compared to long fibers, thus offering greater potential for reducing the insertion loss.