Measure out the frozen fruit puree and chilled ingredients, then assemble everything nearby.
Add the frozen fruit puree, crushed ice, rum, simple syrup, and lime juice into the blender.
Pulse the blender a few times to start breaking down the ice and mix the ingredients loosely.
Blend on high for about 30-60 seconds until the mixture is smooth, frosty, and scoopable, with a slushy consistency. Listen for the smooth whirring sound and check for even blending.
Taste the slushie and add more lime juice or simple syrup if desired, then give it another quick blend to incorporate.
Scoop the icy rum mixture into glasses, garnish with fresh mint leaves, and serve immediately.