A complex-oriented cohomology theory $E^*$ is a multiplicative cohomology theory with a choice of Thom class $x\in\tilde{E}^2(\mathbb{C}P^\infty)$ for the universal complex line bundle (which can be used to define generalised Chern classes for all complex vector bundles).
A real-oriented cohomology theory $F^*$ a multiplicative cohomology theory with a choice of Thom class $x\in \tilde{F}^1(\mathbb{R}P^\infty)$ for the universal real line bundle (which can be used to define generalised Stiefel-Whitney classes for all real vector bundles).
Question 0: Is this correct?
Question 1: Are there any examples of cohomology theories which are both real and complex orientable?
Question 2: (Assuming a yes to Question 1) Are there any results/papers where the interaction between a real and complex orientation is used in an essential way? (I'm thinking perhaps about non-immersion results for real projective spaces.)
Thanks.
Update: Neil's answer and Johannes' comments have answered my original questions: every real oriented cohomology theory is complex oriented, and in fact is a wedge of $H\mathbb{Z}/2$'s. Then let me ask a follow up question. Are there any complex-oriented theories which are not real-orientable but have $E^1(P^\infty)\neq 0$?
