To make the cilantro syrup, add water, sugar, coriander seeds and lime zest to a medium saucepan, and bring to a boil over medium-high heat, stirring until the sugar is dissolved. Take the mixture off the heat and let cool for 15 minutes. Add the cilantro sprigs, cover, and let steep for 4 hours. Strain into a clean bowl and store in the refrigerator until ready to use.
To make the cocktail, add tequila, strawberry puree, cilantro syrup and lime juice to a cocktail shaker with ice. Cap and shake until well-chilled. Strain into a coupe glass and garnish with a cilantro sprig.
* To make strawberry puree give 1 pound (ish) of strawberries a good rinse. Do not dry. Hull and half the strawberries. Quarter any extra large strawberries. Add strawberries to a large saucepan and warm over medium heat, stirring occasionally. Cook until strawberries start to soften and release juices. Remove from heat and transfer to a small food processor. Add a dab of water and process until smooth. Pour mixture through a fine mesh basket strainer into a clean bowl (you can use a spatula here to help press the liquids through strainer). Discard the solids. Let cool before using.