It was inevitable, really... Each fragment of shell has exactly three sharp points, joined by smooth curves. While the King's horses can count reasonably well, his men have been known to confuse donuts and coffee cups, and thus can hardly be trusted with subtle geometric distinctions like adding angles. Between them, they'll eventually manage to label each corner of each fragment with the number of fragments that originally surrounded that point, but if there are any two fragments with identical labels, it's guaranteed that they'll get switched and the reassembly will go poorly. (Even XYZ and XZY will get swapped!) Things could certainly have been worse...all pieces could have an identical set of labels! But is it possible that all fragments' label-sets are distinct, giving HD at least a shot at an open-carton funeral?
Lest this otherwise seem to defy fate, it so happens that the King's llamas and women are able to distinguish concavity from convexity (or at least stickiness from not) well enough to orient a piece labeled XXX or XXY correctly...but is their help sufficient, or for that matter necessary?
(For purposes of this problem, despite occasionally being drawn with what appears to be the beginnings of a digestive tract, you may safely assume that Humpty Dumpty is topologically a sphere)
Extension: Now that it's known to be possible, how few triangles can be used while satisfying the criteria (and how many such solutions exist?) How do these answers change when we eschew the extra help and insist that all three labels are distinct on every triangle?