Whisk the almond milk and arrowroot in a large saucepan until blended. Add the sugar and eggs, cook over medium-low heat, whisking constantly, until the sugar is dissolved and the mixture is slightly thickened, about 8 minutes, taking special care NOT to boil the custard. Remove from the heat and allow the custard to cool slightly.
Whisk the coconut milk into the custard until well blended. Cool, stirring occasionally until cooled to room temp. Refrigerate at least 6 hours.
Stir mixture and process in an ice cream maker according to the manufacturer’s directions. Store leftovers in the freezer.
If avoiding nuts, use rice milk rather than almond milk.