English tomate came from Nahuatl through Spanish. But potato was already present, through Spanish patata, a blend of Taíno batata and Quechua papa; its familiar shape pulled tomate toward tomato. Spanish then contributes tomatillo, literally a “little tomato” despite naming a different fruit.
Sources: Online Etymology Dictionary, tomato; Merriam-Webster, tomato, potato, and tomatillo; RAE, patata, papa, and batata; Academia Mexicana de la Lengua, papa / patata.
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Dates follow the earliest cited evidence for the form and sense shown. Notes name the sources. Timeline spacing is compressed; undated stages are placed in sequence.