If frozen, defrost phyllo.
Preheat the oven to 350 degrees F and brush a 9 x 13-inch baking pan with a little oil or butter.
For the cake, put the eggs, vegetable 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, baking powder, and vanilla. Beat to combine.
Cut phyllo into long, thin ribbons. Add phyllo ribbons 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 peel, water, 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.
When the cake comes out of the oven, pour the syrup on top. Allow cake to cool and absorb all the syrup.
Slice and serve!