In a large, heavy pot, combine the tomatoes, sugar, ginger and cinnamon stick. Set a strainer over the pot and squeeze the lemons into it; discard any seeds. Add the lemon halves to the pot and bring to a boil, stirring gently until the sugar is dissolved. Simmer over low heat, stirring occasionally, until the tomatoes are translucent and very tender, about 1 hour and 15 minutes. Discard the lemons and cinnamon stick.