In a medium mixing bowl, stir the tapioca starch with 1/4 cup of the coconut milk until smooth. Set aside.
Heat the remaining coconut milk in a small saucepan over low heat until it bubbles around the edges.
Whisk the starch mixture into the coconut milk and cook gently over low heat, stirring briskly, until mixture begins to thicken, which will happen quickly. Once it appears to be thickening, remove it immediately from the heat and whisk in the juice.
Transfer the mixture to a heatproof bowl and let it cool to room temperature, then refrigerate until chilled.