Motivated by the study of tunneling-rates and their gauge-dependence, we introduce a distinction between true-vacuum and false-vacuum effective action functionals. This allows to clarify apparent contradictions that arise between the reality and convexity properties of the true-vacuum effective action and the possible existence of false vacua and tunneling processes. Despite the gauge dependence of effective action functionals at zero and finite temperature, we show that tunneling and nucleation rates remain independent of the choice of gauge-fixing. Taking as a starting point the path integral that defines the transition amplitude from a false vacuum onto itself, it is shown that decay rates are exactly determined by a non-convex, false-vacuum effective action evaluated at an extremum. The latter can be viewed as a generalized bounce configuration, and gauge-independence follows from the appropriate Nielsen identities. This holds for any election of gauge-fixing that leads to an invertible Faddeev-Popov matrix. The result, which is nonperturbative and model-independent, not only clarifies issues of convexity, but also how to incorporate radiative corrections in tunneling calculations.