Meyendorff ( Christ in Eastern Christian Thought ) does a good job of showing not only the compatibility of Chalcedon with Cyrillian Christology, but that Chalcedon is essentially Cyrillian. The logic is this:
Chalcedon insisted on a single hypostasis in Christ. The humanity has no separate subsistence, no existence apart from its existence as the humanity of the Word. Chalcedon implies that the human nature is anhypostatic.
Yet, no nature can exist without a hypostasis; without a hypostasis, a nature is just an abstraction. If the human nature of the Son doesn’t have its own hypostasis, then the only possibility is that its hypostasis is the hypostasis of the Word. Chalcedon implies that the human nature is enhypostatic.
If this is so, then the flesh/humanity of the Word is the Word’s own flesh; as Meyendorff suggests, the human nature is after the incarnation as fully the Word’s as His divine nature.
All of Cyril’s theopaschitism, and his insistence on the identity of the hypostasis of the pre-incarnate and incarnate Word, arises as necessary inferences from Chalcedon’s one-hypostasis formula. The Son suffered “in the flesh,” yet it was the Son’s own flesh, not flesh-at-a-distance. He – the Logos – suffered in the humanity He had made His own.
Further, natures don’t die, because natures don’t have a concrete living existence to begin with; the human nature can’t die by itself because it has no hypostasis of its own. Chalcedon thus implies that the Word died in His own flesh.