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We present an inductively coupled-plasma reactive-ion etching process that simultaneously provides both a high etch rate and unprecedented selectivity for gallium phosphide (GaP) in the presence of aluminum gallium phosphide (Al$_\textrm{x}$Ga$_{1-\textrm{x}}$P). Utilizing mixtures of silicon tetrachloride (SiCl$_4$) and sulfur hexafluoride (SF$_6$), selectivities exceeding 2700:1 are achieved at GaP etch rates above 3000 nm/min. A design of experiments has been employed to investigate the influence of the inductively coupled-plasma power, the chamber pressure, the DC bias and the ratio of SiCl$_4$ to SF$_6$. The process enables the use of thin Al$_\textrm{x}$Ga$_{1-\textrm{x}}$P stop layers even at aluminum contents of a few percent.