Nonlinear optical excitation of infrared active lattice vibrations has been shown to melt magnetic or orbital orders and to transform insulators into metals. In cuprates, this technique has been used to remove charge stripes and promote superconductivity, acting in a way opposite to static magnetic fields. Here, we show that excitation of large-amplitude apical oxygen distortions in the cuprate superconductor YBa2Cu3O6.5 promotes highly unconventional electronic properties. Below the superconducting transition temperature (Tc = 50 K) inter-bilayer coherence is transiently enhanced at the expense of intra-bilayer coupling. Strikingly, even above Tc a qualitatively similar effect is observed up to room temperature, with transient inter-bilayer coherence emerging from the incoherent ground state and similar transfer of spectral weight from high to low frequency. These observations are compatible with previous reports of an inhomogeneous normal state that retains important properties of a superconductor, in which light may be melting competing orders or dynamically synchronizing the interlayer phase. The transient redistribution of coherence discussed here could lead to new strategies to enhance superconductivity in steady state.