  We construct a calculus of functors in the spirit of orthogonal calculus,
which is designed to study "functors with reality" such as the Real classifying
space functor, $BU_\mathbb{R}(-)$. The calculus produces a Taylor tower, the
$n$-th layer of which is classified by a spectrum with an action of $C_2
\ltimes U(n)$.
  We further give model categorical considerations, producing a zig-zag of
Quillen equivalences between spectra with an action of $C_2 \ltimes U(n)$ and a
model structure on the category of input functors which captures the homotopy
theory of the $n$-th layer of the Taylor tower.
