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Zinc blende copper iodide is a natural p-type semiconductor that shows promise for transparent electronics, as it combines outstanding transmission of visible light with high hole mobilities. Despite that, remaining challenges still hinder its use in practical applications. To tackle this problem, chalcogen doping has been proposed and tested as a viable approach to enhance hole concentration. To further tune the electronic properties of this transparent semiconductor, we explore here the ternary phase diagram of copper and iodine with sulfur or selenium across the whole composition range. Performing structure prediction calculations using the minima hopping method, we predict 11 stable crystalline ternary structures, 9 of which are unreported. Based on our band structure calculations, we identify among these compounds 4 interesting candidates for p-type transparent materials and we propose alloying of iodine with a chalcogen atom as a way to control the hole concentration. Additionally, we reveal unreported metallic phases with intriguing topological properties. Our findings broaden the scope of potential applications of CuI-based materials, highlighting the possibility of harnessing their unique electronic characteristics in diverse electronic devices and systems.