Make the Coconut Milk Base: In a medium saucepan, whisk together the cornstarch and sugar. Pour in a small amount of the coconut milk and whisk until completely smooth with no lumps remaining. Gradually add the remaining coconut milk, stirring until well combined.
Cook Until Slightly Thickened: Place the saucepan over medium heat and stir frequently until the mixture thickens slightly. This usually takes about 8 to 10 minutes. The mixture should coat the back of a spoon yet remain pourable.
Cool the Mixture: Remove the saucepan from the heat. Cover the surface directly with plastic wrap to prevent a skin from forming. Set aside and allow the mixture to cool.
Blend the Strawberries: Add the strawberries to a blender and blend until completely smooth, creating a silky puree.
Make the Strawberry Layer: Pour one-third of the cooled coconut milk mixture into the strawberry puree. Blend again until smooth and evenly combined.
Make the Matcha Layer: Add the matcha powder to the remaining two-thirds of the coconut milk mixture. Whisk until the matcha is fully incorporated and no lumps remain.
Fill the Popsicle Molds: Alternately pour the mixture into popsicle molds, layering the matcha mixture and strawberry mixture, beginning and ending with the matcha layer. For a more decorative layered effect, slowly pour each mixture along the inside wall of the mold.
Remove Air Bubbles: Gently tap the filled molds on the countertop a few times to release any trapped air bubbles.
Insert the Popsicle Sticks: Place the mold cover on top if your popsicle mold includes one. Insert a popsicle stick into the center of each mold, leaving about 1½ inches (4 cm) exposed.
Freeze: Transfer the molds to the freezer and freeze for at least 8 hours, or until completely firm.
Unmold and Serve: When ready to serve, fill a shallow pan with hot water. Dip the bottom of the popsicle mold into the hot water for about 10 seconds to help loosen the popsicles. Gently pull them out and enjoy.