Heat the coconut milk and sweetener over medium heat until chosen sweetener is dissolved and mixture is hot. Do not let this boil!
Remove from heat and temper in the egg yolks. To do this, beat the egg yolks in a blender or in a medium bowl with a whisk until smooth. Blending/whisking constantly, slowly add a small amount of the hot coconut milk mixture at a time until completely incorporated and smooth.
Return the sweetened milk and egg mixture to pan, add the salt, and simmer for 2-3 minutes or until slightly thickened, stirring occasionally.
Meanwhile, puree the fresh strawberries in the blender until smooth.
Remove the coconut milk mixture from the heat, stir in the vanilla, and add mixture to the blender with the strawberries. Blend until mixed and completely smooth.
Let mixture cool in the refrigerator for approximately 1 hour.
Pour into ice cream maker (following instructions) or use tin can method. Freeze for 1 hour before serving.
If you aren't an ice cream purist and trust the source of your eggs, you can simply blend all the ingredients in a blender until smooth and add to the ice cream maker, skipping all the other steps. This will result in a slightly less creamy (but equally delicious) ice cream.