  Lefschetz thimbles have been proposed recently as a possible solution to the
complex action problem (sign problem) in Monte Carlo simulations. Here we
discuss pure abelian gauge theory with a complex coupling $\beta$ and apply the
concept of Generalized Lefschetz thimbles. We propose to simulate the theory on
the union of the tangential manifolds to the thimbles. We construct a local
Metropolis-type algorithm, that is constrained to a specific tangential
manifold. We also discuss how, starting from this result, successive subleading
tangential manifolds can be taken into account via a reweighting approach. We
demonstrate the algorithm on $U(1)$ gauge theory in 1+1 dimensions and
investigate the residual sign problem.
