If you are looking for Falernum, the wine, see Falerno.
Velvet Falernum (the full name) is a clear liqueur made in the Barbados, where it is very popular. It is made from lime, cane syrup, ginger, and flavourings including almonds and cloves. The liqueur ends up 11.5% alcohol.
Falernum can be blended with rum, drunk straight up as a liqueur, or used as a flavouring in desserts with fruit.
There is now a non-alcoholic version being made as a clear flavouring syrup called Falernum Syrup.
To make your own:
1 cup (8 oz / 250 ml) white rum
zest of 3 limes
3 drops Almond Extract
4 cups (32 oz / 450g) sugar
4 cups (32 oz / 1 litre) of water
Place all ingredients except water and sugar in a covered container, and let steep for 24 hours.
Mix sugar and water, place in a separate covered container for 24 hours, shake occasionally to ensure it dissolves completely.
Strain rum mixture into the sugar syrup. If desired, you can add another few drops of almond extract to accentuate the almond flavour, which should be detectable, but barely so.
Makes 5 cups / 40 oz / 1.2 litres
Some people use Grenadine as a substitute. Though Grenadine, which is coloured, is quite different from the clear liqueur, the taste is similar to the Falernum Syrup.
Developed in Bridgetown, Barbados by a John D. Taylor in 1890.