We present a triple point of a new kind for general relativity, at which two gravitational solitons can coexist with a planar black hole in anti-de Sitter space. Working in the context of nonlinear electrodynamics, we obtain simple, sensible spacetimes for which the thermodynamics can be studied in an analytic manner. The spacetimes are charged under the nonlinear electrodynamics leading to an electric charge for black holes and a magnetic flux for solitons. In the grand-canonical ensemble, we show that the phase space of the theory is very rich, containing reentrant phase transitions, as well as triple points, for small values of the coupling controlling the nonlinearity of the electrodynamics Lagrangian.