Why are real orbitals used to define the orbital of hydrogenic atoms instead of complex orbitals?
The main disadvantage of complex atomic orbitals (AOs) is that many of them are hard to visualize.  For example, try making a simple sketch explaining how $p_{-1}$ differs from $p_{1}$. Conversely the complex AOs have very few  advantages over real MOs - they are all eigenfunctions of $\hat{L}_z$ but that's not really important for molecular  and closed shell systems. 

So with many key advantages and very few disadvantages real orbitals are understandably the most popular choice.