Aren’t real vector bundles just like complex vector bundles?  Well, yes, and KO*(X) is pretty similar formally to KU*(X) (except 2 is not equal to 8) . Generalize K-theory to real and to complex C*- algebras, though, and you see evidence of the secret life. Teaser: every commutative complex C*-algebra is isomorphic to C_o(X,C) for some locally compact X. It’s false for real C*-algebras and the example appears above: it is the complex numbers C, thought of as a real commutative C*-algebra of dimension 2.  It is not isomorphic to R^2 even as a ring!