Epitaxial strain alters the physical properties of thin films grown on single crystal substrates. Thin film oxides are particularly apt for strain engineering new functionalities in ferroic materials. In the case of La(2/3)Ca(1/3)MnO(3) (LCMO) thin films, here we show the first experimental images obtained by electron holography demonstrating that epitaxial strain induces the segregation of a flat and uniform nonferromagnetic layer with antiferromagnetic (AFM) character at the top surface of a ferromagnetic (FM) layer, the whole film being chemical and structurally homogeneous at room temperature. For different substrates and growth conditions the tetragonality of LCMO at room temperature, defined as τ = |c - a|/a, is the driving force for a phase coexistence above an approximate critical value of τC ≈ 0.024. Theoretical calculations prove that the increased tetragonality changes the energy balance of the FM and AFM ground states in strained LCMO, enabling the formation of magnetically inhomogeneous states. This work gives the key evidence that opens a new route to synthesize strain-induced exchanged-biased FM-AFM bilayers in single thin films, which could serve as building blocks of future spintronic devices.