Even though quantum theory uses complex Hilbert spaces and they play a key "tidying" role in the theory, it is only fairly recently that physicists have started to ask if the quantum world is inherently complex. Very recently a Bell-like experiment based on a network scenario is proposed that numerically separates complex from real quantum theory. In brief, it has now been shown that the real-world is not! In this talk we discuss the similarities/differences between the real and complex case for various concepts like, entanglement and separability, positive versus completely positive maps and the various characterizations of entanglement breaking maps. We also discuss the real version of the PPT-squared conjecture. This is a joint work with Giulio Chiribella, Ken Davidson and Vern Paulsen.