Preheat the oven to 350 degrees F and brush a 9 x 13-inch baking pan with olive oil.
For the cake, put the eggs, olive oil, and sugar in a large bowl. Beat the mixture well with an electric mixer until the ingredients are well combined. Add the yogurt, orange zest, and baking powder, and beat to combine.
Crush the phyllo sheets with your hands. Add the crushed phyllo to the cake batter and stir it in with a spatula. Evenly spread the batter in the prepared pan.
Bake the cake for 45 to 55 minutes, until a knife inserted into the center comes out clean.
Make the syrup while the cake is baking. In a medium saucepan, stir together the orange juice and sugar. Bring the mixture to a boil then simmer it over medium heat for 20 minutes, until it thickens to the consistency of maple syrup, stirring often. Let the syrup cool.
Slice the cake into 20 pieces and leave it to cool. When the cake and syrup are cool, pour all of the syrup on top of the cake; it will be absorbed.