Pour half-and-half into a medium saucepan and place over medium heat. Add sugar and stir until dissolved. Continue to heat until half-and-half is hot, but do not let it come to a boil.
Whisk egg yolks in a small bowl until pale yellow.
Slowly add about 1 cup of the half-and-half mixture to the egg yolks, whisking constantly. Repeat with another cup. Pour the egg yolk mixture into the saucepan with the remaining half-and-half mixture and cook over medium heat until the mixture thickens and coats the back of a spoon. Do not let the mixture boil.
Remove from heat and stir in spearmint and coconut extract. Pour mixture into a bowl with a pour spout and let cool for 30 minutes. Refrigerate, 8 hours to overnight.
Pour 1/2 of the mixture into an ice cream maker and churn to the consistency of soft-serve. Add strawberry puree. Continue to churn until ice cream has thickened. Transfer ice cream to a bowl and clean out the ice cream maker with a rubber spatula. Place strawberry ice cream in the freezer.
Pour the remaining ice cream base into the ice cream maker and churn to the consistency of soft-serve. Add mango puree. Continue to churn until the ice cream has thickened.
Remove strawberry ice cream from the freezer. Place a large scoop of strawberry ice cream in a 9-inch loaf pan using 2 large spoons. Top with a large scoop of mango ice cream. Fill up the loaf pan, continuing to alternate flavors. Cover with plastic wrap and place into the freezer for a minimum of 4 hours.