1 Long Tom: Dealing wi' these British sailors be makin' me thirsty. Dirque! Be mixin' me up a daiquiri!
2 Dirque: Aye aye, cap'n!
3 Dirque: Oops!
3 British Sailor 2: Look out, you clumsy oaf! You've spilt lime juice all over us!
4 Long Tom: Arrr! Be takin' these Limey Jack Tars away!

A daiquiri is a cocktail made primarily of rum and lime juice.

I came up with the joke first, and then (not being familiar with daiquiris) searched for a cocktail made with rum (since it's the pirates' drink of choice) and lime juice (to set up the "limey" pun). I was very pleased to find that this is essentially the description of a classic daiquiri, the main difference being merely the addition of simple sugar syrup to reduce the sourness and give it a bit more sweetness. There's also the mojito, which is also rum and lime juice and sugar syrup, but muddled with mint leaves and topped with soda water, so that seemed less pure. And besides "daiquiri" is an inherently funnier word than "mojito".

By the way, British sailors, and later by extension all British people, are colloquially referred to as limeys because of the practice of the Royal Navy of adding citrus juice (either lemon or lime) to the sailors' daily ration of grog (watered down rum), as a means of preventing scurvy. This began in the mid-18th century, so about a hundred years after the setting for this comic. The daiquiri was only formalised as cocktail after 1900, but it is similar in composition to the grog and lime juice used by British sailors.