Added to the example “real vector bundles as Real vector bundles” (here) a line saying that the equivalence continues to hold over varying base spaces.

Then below that I added a Remark (here) that, in contrast, complex vector bundles over varying bases do $\,$ not $\,$ form a full subcategory of Real vector bundles over varing $\mathbb{Z}/2$-spaces — but they do form a full subcategory of “Real vector bundles equipped with complex structure”, where “complex structure” means endomorphisms (of Real vector bundles) squaring to $-1$.

In this sense, the term “Real vector bundle” is quite apt.

]]>started a subsection (here) on Atiyah Real vector bundles

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