Cut the papayas into large dice.
Finely grate the rind of both limes, then squeeze 1 of the limes and reserve the juice. Cut the pith off the second lime and segment the flesh over the bowl of diced papaya to catch the juice. Add the lime segments and grated rind to the papaya.
Pour the lime juice into a small saucepan with the sugar and heat gently until the sugar has dissolved. Remove from the heat and leave to cool.
Pour the cooled lime juice over the fruit and toss thoroughly. Add the toasted almonds and serve with lime wedges.