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The low-frequency linearly-polarised radio source population is largely unexplored. However, a renaissance in low-frequency polarimetry has been enabled by pathfinder and precursor instruments for the Square Kilometre Array. In this second paper from the POlarised GaLactic and Extragalactic All-Sky Murchison Widefield Array (MWA) Survey -- the POlarised GLEAM Survey, or POGS -- we present the results from our all-sky MWA Phase I Faraday Rotation Measure survey. Our survey covers nearly the entire Southern sky in the Declination range $-82^{\circ}$ to $+30^{\circ}$ at a resolution between around three and seven arcminutes (depending on Declination) using data in the frequency range 169$-$231 MHz. We have performed two targeted searches: the first covering 25,489 square degrees of sky, searching for extragalactic polarised sources; the second covering the entire sky South of Declination $+30^{\circ}$, searching for known pulsars. We detect a total of 517 sources with 200 MHz linearly-polarised flux densities between 9.9 mJy and 1.7 Jy, of which 33 are known radio pulsars. All sources in our catalogues have Faraday rotation measures in the range $-328.07$ rad m$^{-2}$ to $+279.62$ rad m$^{-2}$. The Faraday rotation measures are broadly consistent with results from higher-frequency surveys, but with typically more than an order of magnitude improvement in the precision, highlighting the power of low-frequency polarisation surveys to accurately study Galactic and extragalactic magnetic fields. We discuss the properties of our extragalactic and known-pulsar source population, how the sky distribution relates to Galactic features, and identify a handful of new pulsar candidates among our nominally extragalactic source population.