Remove the zest of 4 limes with a vegetable peeler or zester, being careful to avoid the white pith.
Squeeze the limes to make 1/2 cup of juice and set the juice aside.
Put the zest in a food processor fitted with a steel blade.
Add the sugar and process for 2 to 3 minutes, until the zest is very finely minced.
In the bowl of an electric mixer fitted with a paddle attachment, cream the butter with the sugar and lime zest.
Add-- the eggs, 1 at a time, and then add the lime juice and salt and rum.
Mix until combined.
Pour the mixture into a 2-quart saucepan and cook over low heat, stirring constantly, until thickened, about 10 minutes.
The lime curd will thicken at about 175 degrees F, or just below a simmer.
Remove from the heat and set aside.
Fill the prepared tart shell with warm lime curd and allow to set at room temperature.
Cover Lightly and refrigerate until ready to serve.
Garnish with thin twists of lime slices and zest, add a sprig of peppermint.