8 to 12 ounces firm, salty cheese, such as queso fresco, feta, or ricotta salata, cut into 1/4-inch slices
Bring a large pot of water to a boil. Have ready, a bowl of ice water and a spider or slotted spoon. With a paring knife, cut the stems out of the tomatoes and cut an X in the opposite end. Slip the tomatoes into the boiling water and leave them until the skin around the X starts to loosen, 30 seconds to 1 minute. With the spider, scoop the tomatoes into the ice water. When cool enough to handle, slip the skins off the tomatoes and cut the tomatoes in half lengthwise. Scoop out and discard the seeds. Set the seeded tomato halves aside.
Bring 3 cups water, the sugar, cinnamon, and allspice to a boil in a medium saucepan over high heat. Adjust the heat to the lowest setting and let steep for 20 minutes.
Add the tomatoes, and continue to simmer on low heat until the syrup has thickened about another 20 minutes. Cool, then chill. Remove allspice berries and cinnamon sticks. Serve the tomatoes in a compote dish or small bowl, accompanied by the sliced cheese.