Combine the first 3 ingredients in a medium bowl, and stir with a whisk.
Cook milk in a heavy saucepan over medium-high heat to 180° or until tiny bubbles form around the edge (do not boil). Gradually add hot milk to egg mixture, stirring constantly with a whisk. Return milk mixture to pan; cook over medium heat until thick (about 8 minutes), stirring constantly. Remove from heat.
Place pan in a large ice-filled bowl until custard cools to room temperature (about 15 minutes), stirring frequently. Stir in the yogurt, rum, and vanilla; pour mixture into a glass bowl. Cover and place in freezer 8 hours. Remove mixture from freezer; let stand 15 minutes. Place in a food processor; process until smooth. Serve immediately.