In food processor, combine strawberry and mango chunks. Pulse a few times to break down fruit. The mixture will appear grainy and in smaller pieces. Continue to blend, scraping down sides of bowl occasionally, until mixture starts to soften and combine, about 3 minutes. When mixture resembles soft-serve ice cream, stop and add lime juice, maple syrup, and chopped basil. Process until all ingredients are blended. Check for sweetness and adjust to taste.