Preheat the oven to 350 degrees F (175 degrees C). Grease and flour a loaf pan or line with parchment paper.

Combine sugar, 2 tablespoons of Mojito spice, 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

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.

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.

Cool in the pan for 5 to 10 minutes before inverting onto a cooling rack set over a baking sheet.

Prepare the glaze while the loaf is cooling. Whisk powdered sugar, lime juice, and Mojito Spice in a small saucepan. Cook and stir over low heat until glaze is smooth. Remove from heat and stir in rum until completely combined.

Strain glaze into a measuring cup with a spout. Let sit until slightly thickened, about 5 minutes.

Pour the glaze over the cooling cake, letting excess drip down the sides. Let loaf cool and glaze firm up before slicing.