The Chalcedonean Definition is the official Christology of the Catholic Church. It states that Christ is a single person with two natures: human and divine. These two natures are neither divided nor mixed.
The Oriental Orthodox Christological position is Miaphysitism. This states that Christ has a single nature, but that nature is both fully human and fully divine, with the human and divine elements being neither mixed with nor divided from each other.
Historically the Catholics have misunderstood miaphysitism as a form of monophysitism (Christ has a single nature which is either human or divine but not both) and the Oriental Orthodox have misunderstood Chalcedonean definition as being a form of Nestorianism (Christ was two persons).
In my analysis, both sides are going to great pains to ensure that Christ is fully human and fully divine. They both go about it in different ways, but both succeed in the goal, and so both are acceptable formulations
Recently the two sides have cooled off and the accusations of heresy against each other are not so strong. There have been moves by some to say that the dispute is just one of terminology, not substance. This includes joint statements signed by Pope Paul VI and Pope John Paul II, but despite this, the Catholic and Oriental orthodox Churches have not officially declared themselves in communion with each other, so I am not sure if Miaphysitism is considered an acceptable Christological position by the Catholic church.
Personally I find Miaphysitism to be more coherent and intuitive than the Chalcedonean 2 natures definition. As a Catholic, am I permitted to believe Miaphysitism so long as I don't simultaneously deny Chalcedon? And if it is permitted, then why aren't the Oriental Orthodox churches in full communion with the Catholic Churches as the Eastern Orthodox churches are?
Related question: is this what the Eastern Catholic churches that have an Oriental Orthodox background do? Ie. Accept that the Chalcedonean definition is Orthodox, but go ahead and teach Miaphysitism anyway? (I'm thinking of the Ethiopian, Eritrean and Coptic Catholic churches for example)
edit: I note that the wikipedia article for Miaphysitism says the following:
Historically, Chalcedonian Christians have considered Miaphysitism in general to be amenable to an orthodox interpretation, but they have nevertheless perceived the Christology of the Oriental Orthodox to be a form of Monophysitism (single nature doctrine).
This seems to indicate that Chalcedonians are able to accept Miaphysitism, given an appropriate interpretation.