We study a recently proposed formulation of U(1) lattice field theory with electric and magnetic
matter based on the Villain formulation. This discretization allows for a duality that gives rise to
relations between weak and strong gauge coupling. There exists a self-dual value of the gauge
coupling where one may study the model as a function of the remaining matter coupling. Using
Monte Carlo simulations based on a worldline/worldsheet representation of the system we evaluate order parameters for spontaneous breaking of self-duality. We find that in some interval of the
matter coupling self-duality becomes broken spontaneously. We determine the endpoints of this
interval and study the nature of the corresponding critical points. Finally we explore the system
away from the self-dual gauge coupling and show that when crossing the self-dual point a first
order jump is seen in the order parameters.