In a medium saucepan, on medium heat, stir together cream, milk, and ½ cup of sugar. Heat until warm, barely a bubble, and not boiling.
In a large bowl, hand whisk together egg yolks, cornstarch, and remaining ½ cup of sugar. Whisk until light in color, about two minutes.
Slowly pour half of the hot mixture into the egg mixture, continually whisking.
Pour the egg and cream mixture (tempered eggs) back into saucepan with remaining cream mixture. Stir gently with a wooden spoon for 2-4 minutes, until thick enough to coat the back of the spoon.
Remove from heat and stir in the Banana rum Cream, and vanilla. Pour mixture through a sieve into a large bowl.
Cover bowl with plastic wrap and refrigerate for 4 hours, until thoroughly chilled.
Transfer sweet ice cream base into the ice cream maker and churn according to manufacturer's instructions. It should thicken and resemble soft serve ice cream. (about 25-30 minutes)
Once thickened fold in the banana chunks and transfer ice cream to an airtight container and freeze for 5-6 hrs or overnight.