1. Make tomato ginger water: Purée one fourth of tomatoes with ginger in a food processor until smooth, about 2 minutes. Transfer to a small heavy saucepan and simmer, covered, stirring occasionally, 5 minutes.
2. Clean processor thoroughly, then purée remaining tomatoes in batches with chiles, salt, and sugar until smooth, 2 to 3 minutes. Transfer to a bowl and stir in cooked tomato ginger purée.
3. Line a large sieve with stacked cheesecloth squares and set over a deep nonreactive pot or a large clean bucket. Carefully pour purée into center of cheesecloth. Gather sides of cheesecloth up over purée to form a large sack and tie sides together securely with string as close to purée as possible but without squeezing purée. Tie sack to a wooden spoon longer than diameter of pot and remove sieve. Set spoon across top of pot, suspending sack inside pot and leaving a couple of inches underneath sack so that it will not sit in any tomato water that accumulates. (Alternatively, transfer tomato water to another container as it accumulates.) Let sack hang 4 hours at room temperature.
4. Prepare clams: Bring 1 cup water to a boil in a 7- to 8-quart heavy pot. Add clams and cook, covered, over moderately high heat, until clams are opened wide, stirring and checking frequently after 6 minutes and transferring clams as opened to a bowl. (Discard any clams that have not opened after 10 minutes.) When cool enough to handle, shuck clams.
5. Assemble shooters: Discard cheesecloth sack and its contents without squeezing. Pour tomato water through large sieve lined with a dampened paper towel into a bowl. Measure out 4 cups tomato water and reserve remainder for another use.
6. Transfer 1/2 cup tomato water to a small saucepan, then sprinkle with gelatin and let soften 1 minute. Heat over low heat, stirring, just until gelatin is dissolved, then stir into remaining 31/2 cups tomato water.
7. Arrange glasses on trays and put 1 or 2 cilantro leaves in each, then a clam, and then 1 or 2 more cilantro leaves. Add about 1 tablespoon tomato water to each glass (stir water each time so gelatin remains well distributed) and chill, loosely covered with plastic wrap, at least 3 hours. (Gelatin will be thickened but not set; it needs to slide easily out of glass.)
8. Cooks' notes: ·Tomato water (without gelatin) can be made 2 days ahead. Pour through paper-towel-lined sieve, then chill, covered. ·Shooters can be chilled up to 1 day.