Asynchronous variational integration of layered contact models provides a framework for robust collision handling, correct physical behavior, and guaranteed eventual resolution of even the most difficult contact problems. Yet, even for low-contact scenarios, this approach is significantly slower compared to its less robust alternatives-often due to handling of stiff elastic forces in an explicit framework. We propose a method that retains the guarantees, but allows for variational implicit integration of some of the forces, while maintaining asynchronous integration needed for contact handling. Our method uses phantom meshes for calculations with stiff forces, which are then coupled to the original mesh through constraints. We use the augmented discrete Lagrangian of the constrained system to derive a variational integrator with the desired conservation properties.