  Let $G_\mathbb R$ be a connected real reductive group and let $X$ be the
corresponding complex symmetric variety under the Cartan bijection. We
construct a canonical equivalence between the relative Satake category of
$G(\mathcal O)$-equivariant $\mathbb C$-constructible complexes on the loop
space of $X$ and the real Satake category of $G_\mathbb R(\mathcal O_\mathbb
R)$-equivariant $\mathbb C$-constructible complexes on the real affine
Grassmannian. We show that the equivalence is $t$-exact with respect to the
natural perverse $t$-structures and is compatible with the fusion products and
Hecke actions. We further show that the relative Satake category is equivalent
to the category of $\mathbb C$-constructible complexes on the moduli stack of
$G_\mathbb R$-bundles on the real projective line $\mathbb P^1(\mathbb R)$ and
hence provides a connection between the relative Langlands program and the
geometric Langlands program for real groups. We provide numerous applications
of the main theorems to real and relative Langlands duality including the
formality and commutativity conjectures for the real and relative Satake
categories and an identification of the dual groups for $G_\mathbb R$ and $X$.
