Melt the butter in a saucepan and add the flour. Cook, stirring, for 1-2 minutes.
Remove from the heat and add the eggs, mashing them to a smooth paste. Add the juice and rind or the lime, the fish stock, milk, cream and the crabmeat.
Return the saucepan to the heat and simmer very gently for 10 minutes, stirring from time to time.
Season with salt and pepper, add the bitters and sherry and reheat gently. Serve immediately.