  We consider the extension of classical 2-dimensional topological quantum
field theories to Klein topological quantum field theories which allow
unorientable surfaces. We approach this using the theory of modular operads by
introducing a new operad governing associative algebras with involution. This
operad is Koszul and we identify the dual dg operad governing A-infinity
algebras with involution in terms of Mobius graphs which are a generalisation
of ribbon graphs. We then generalise open topological conformal field theories
to open Klein topological conformal field theories and give a generators and
relations description of the open KTCFT operad. We deduce an analogue of the
ribbon graph decomposition of the moduli spaces of Riemann surfaces: a Mobius
graph decomposition of the moduli spaces of Klein surfaces (real algebraic
curves). The Mobius graph complex then computes the homology of these moduli
spaces. We also obtain a different graph complex computing the homology of the
moduli spaces of admissible stable symmetric Riemann surfaces which are partial
compactifications of the moduli spaces of Klein surfaces.
