How can we classify a bifurcating process consistently (ie, without contradiction)

The difference between Linnean systematics (ie, evolutionary taxonomy) and cladistic classification is just which of them that is a consistent (ie, non-contradictory) classification of a bifurcating process:

Linnean systematics assumes that the entities of the process are real and the class abstract, whereas cladistic classification assumes that the class of the process is real and the objects abstract. It means that whereas Linnean systematics consistently keeps the concepts "entitity", "finite class" and "infinite class" apart, cladistic classification instead conflates these concepts by using "a single clade" as entity, the class "biological species" as finite class and the class "clade" as infinite class. What Linnean systematics thus keeps consistently apart do cladistic classification instead conflate.

It means that if we want to classify a bifurcating process consistently (ie, without contradiction), we have to use an orthogonal system like the Linnean systematics, and that cladistic classification takes us back to where we started from (ie, to the contradiction we thus avoided by the orthogonal system).

The answer is thus that it is Linnean systematics (ie, evolutionary taxonomy) that is a consistent (ie, non-contradictory) classification of a bifurcating process. Cladistic classification is instead an inconsistent (ie, contradictory) classification of a bifurcating process (yet consistently inconsistent).