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Explanation: 'avant-dire-droit' judgment is a judgment given during the proceedings. In England the closest translation would be an interim order.
HOWEVER, in EU law there is such a thing as an 'interim judgment'. Since EU law is closer to French law than to English law, I would use that expression. Also, it seems that a French 'interim judgment' is delivered by a judge other than the judge who will give the final judgment, and the latter need not take the interim judgment into consideration (http://juripole.u-nancy.fr/Dictionnaire/Italien/F.html).
Execution avant-dire-droit would be execution of the interim judgment.