Combine the coconut milk and the cream in a medium saucepan over low heat. Heat just until bubbles form around the edges. Remove from heat. In a large bowl, beat the yolks, sugar, and syrup until pale yellow. Slowly add the hot milk mixture while stirring. Transfer to a clean saucepan and gently heat, stirring constantly, until the mixture coats the back of a spoon. Transfer to a clean bowl and stir in the ginger and the lemon juice. Chill for 1 hour. Freeze in an ice cream maker according to manufacturer's instructions.