The real \(\mathrm{K}\)-theory spectrum \(\mathrm{KO}\) is “almost complex oriented’’. Here are a collection of properties that demonstrate this:

(1) \(\mathrm{KO}\) is the \(C_2\) fixed points of a complex oriented cohomology theory \(\mathrm{KU}\).

(2) Complex oriented cohomology theories have trivial Hurewicz image, whereas \(\mathrm{KO}\) has a small Hurewicz image – it detects \(\eta\) and \(\eta^2\).

(3) Complex oriented cohomology theories receive a ring map from \(\mathrm{MU}\). \(\mathrm{KO}\) receives no ring map from \(\mathrm{MU}\) but it receives one from \(\mathrm{MSU}\).

(4) If \(E\) is a complex orientable cohomology theory, every complex vector bundle \(V\) is \(E\)-orientable. Not every complex vector bundle \(V\) is \(\mathrm{KO}\)-orientable, but \(V\oplus V\) and \(V^{\otimes 2}\) are.

Higher real \(E\) theory \(\mathrm{EO}\) is an odd primary analogue of \(\mathrm{KO}\). At \(p=3\), \(\mathrm{EO}\) is closely related to \(\mathrm{TMF}\). \(\mathrm{EO}\) is defined as the \(C_p\) fixed points of a complex oriented cohomology theory, and it has a small but nontrivial Hurewicz image, so it satisfies analogues of properties (1) and (2). I prove that it also satisfies analogues of properties (3) and (4). In particular, I produce a unital orientation map from a Thom spectrum to \(\mathrm{EO}\) and prove that for any complex vector bundle \(V\) the bundles \(pV\) and \(V^{\otimes p}\) are complex oriented.