  We define an extended field theory in dimensions $1+1+1$, that takes the form
of a `quasi 2-functor' with values in a strict 2-category
$\widehat{\mathcal{H}am}$, defined as the `completion of a partial 2-category'
$\mathcal{H}am$, notions which we define. Our construction extends Wehrheim and
Woodward's Floer Field theory, and is inspired by Manolescu and Woodward's
construction of symplectic instanton homology. It can be seen, in dimensions
$1+1$, as a real analog of a construction by Moore and Tachikawa.
  Our construction is motivated by instanton gauge theory in dimensions 3 and
4: we expect to promote $\widehat{\mathcal{H}am}$ to a (sort of) 3-category via
equivariant Lagrangian Floer homology, and extend our quasi 2-functor to
dimension 4, via equivariant analogues of Donaldson polynomials.
