Step 1
Preheat the oven to 350 degrees F (175 degrees C). Grease and flour a loaf pan or line with parchment paper.
Step 2
Combine sugar lime zest and mint in a medium bowl. Rub together with your fingers until moist and fragrant. Add lime juice and rum; whisk until dissolved. Whisk in yogurt oil and eggs.
Step 3
Stir flour baking powder and salt together in a large bowl. Slowly add the wet ingredients mixing until smooth. Pour batter into the prepared pan.
Step 4
Bake in the preheated oven until golden brown and a toothpick inserted into the center comes out clean about 50 minutes. Tent loaf with foil if it starts browning too quickly.
Step 5
Cool in the pan for 5 to 10 minutes before inverting onto a cooling rack set over a baking sheet.
Step 6
Prepare the glaze while the loaf is cooling. Whisk powdered sugar lime juice and mint leaves in a small saucepan. Cook and stir over low heat until glaze is smooth and mint has wilted about 5 minutes. Remove from heat and stir in rum until completely combined.
Step 7
Strain glaze into a measuring cup with a spout pressing against the mint leaves to extract as much flavor as possible. Let sit until slightly thickened about 5 minutes.
Step 8
Pour the glaze over the cooling cake letting excess drip down the sides. Let loaf cool and glaze firm up before slicing.