The Council of Chalcedon of 451 gave a clear definition of the Hypostatic Union, the doctrine that in the incarnation, the Son of God took on a full human nature, so that in this one person are united two natures, divine and human. Hypostasis refers to one of the three persons of God, so the Hypostatic Union means that the union between divine and human takes place in the person of God the Son.
Following, then, the holy Fathers, we all unanimously teach that our Lord Jesus Christ is to us One and the same Son, the Self-same Perfect in Godhead, the Self-same Perfect in Manhood; truly God and truly Man; the Self-same of a rational soul and body; co-essential with the Father according to the Godhead, the Self-same co-essential with us according to the Manhood; like us in all things, sin apart; before the ages begotten of the Father as to the Godhead, but in the last days, the Self-same, for us and for our salvation (born) of Mary the Virgin Theotokos as to the Manhood; One and the Same Christ, Son, Lord, Only-begotten; acknowledged in Two Natures unconfusedly, unchangeably, indivisibly, inseparably; the difference of the Natures being in no way removed because of the Union, but rather the properties of each Nature being preserved, and (both) concurring into One Person and One Hypostasis; not as though He was parted or divided into Two Persons, but One and the Self-same Son and Only-begotten God, Word, Lord, Jesus Christ; even as from the beginning the prophets have taught concerning Him, and as the Lord Jesus Christ Himself hath taught us, and as the Symbol of the Fathers hath handed down to us.
The Chalcedonian Definition was written to refute several positions considered heretical: that Christ is not of the same nature as the Father (Arianism), that he does not have a full human nature (Apollinarism), that Christ was a fusion of divine and human (Eutyches), that the two natures of Christ were not joined in one person (Nestorianism).
This Chalcedonian theology was not accepted by all churches who attended the council, and to this day is rejected by the Oriental Orthodox churches. The Oriental Orthodox churches too say that they reject the four positions above, but rather than saying that Christ has two natures, they teach Miaphysitism, that
in the one person of Jesus Christ, Divinity and Humanity are united in one "nature" ("physis"), the two being united without separation, without confusion, and without alteration. (orthodoxwiki.org)
This sounds on the surface quite similar to the position of Eutyches, but the Oriental Orthodox are adamant that they too reject his teaching.
While there have been some moves towards declaring the dispute between Miaphysitism and Chalcedonian a difference purely of terminology and not of substance, support for this has been limited on both sides, with many strongly opposed to saying that the dispute has been resolved.
So for those Chalcedonian theologians who do reject that Miaphysitism is compatible with the Christology taught in the Chalcedonian Definition, what are the major problems they see with Miaphysitism?