Place a large bowl in an ice-water bath. Set aside.
In a medium bowl, whisk together egg yolks and ½ cup sugar.
In a medium saucepan, bring cream, remaining ¼ cup sugar, and salt just to a boil over medium heat. Whisk approximately one-third of hot cream mixture into egg mixture in a slow steady stream. Transfer egg mixture to saucepan; cook, stirring constantly, until a candy thermometer registers 180°, about 1 minute.
Remove pan from heat. Stir in cream cheese until well combined. Strain mixture through a fine-mesh sieve into prepared bowl in ice-water bath. Stir in banana, lime juice, and vanilla extract. Let mixture cool until chilled, stirring occasionally. Remove bowl from ice water bath; cover with plastic wrap, and refrigerate for at least 3 hours.
Freeze mixture in an ice-cream maker according to manufacturer’s instructions. (Ice cream will be soft.) Transfer to a resealable container. Stir in salted caramel sauce, smoothing top with a spoon. Freeze until firm, approximately 4 hours.
Recipe by Southern Lady Magazine at https://southernladymagazine.com/banana-split-recipe/