Combine garlic and 1 cup oil in a small saucepan over medium-low heat. Cook 5 minutes or until garlic starts to sizzle. Remove from heat; let cool completely.
Combine vinegar, salt, and egg yolks in a blender. Process until combined. Combine garlic oil and remaining 2 cups oil in a 4-cup glass measuring cup. With blender running, slowly drizzle oil mixture into blender; process 2 minutes. Transfer mixture to a large bowl; stir in rum.