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Highly siderophile elements (HSEs) are strongly depleted in the bulk silicate Earth (BSE) but are present in near-chondritic relative abundances. The conventional explanation is that the HSEs were stripped from the mantle by the segregation of metal during core formation but were added back in near-chondritic proportions by late accretion, after core formation had ceased. Here we show that metal-silicate equilibration and segregation during Earth's core formation actually increased HSE mantle concentrations because HSE partition coefficients are relatively low at the high pressures of core formation within Earth. The pervasive exsolution and segregation of iron sulfide liquid from silicate liquid (the "Hadean matte") stripped magma oceans of HSEs during cooling and crystallization, before late accretion, and resulted in slightly suprachondritic palladium/iridium and ruthenium/iridium ratios.