Can you have Identical twins with two different genders?

These are one of the questions that make me facepalm throughout my day and throughout the night because it is not possible.

Identical twins, or monozygotic twins, are babies that come from the same egg. The same cell. Because of this, they have exactly the same DNA.

*This is about normal situations, abnormal situations would result in one of the babies getting Turners syndrome or have some unfortunate congenital condition happen to them.