The problem is in allowing perturbations around effective w_{DE}=-1. The "phantom divide crossing" is the evolution of dark energy's effEOS across w = -1, the

The problem is in allowing perturbations around effective w_{DE}=-1. The "phantom divide crossing" is the evolution of dark energy's effEOS across w = -1, the boundary between a quintessence regime (w > -1) and a phantom dark energy (w < -1) regime. Phantom models generically violate the null energy condition. A local crossing thus causes all sorts of problems for minimally coupled single scalar field DE (see e.g. https://doi.org/10.1103/PhysRevD.78.087303 https://doi.org/10.1103/PhysRevD.78.087303 aka https://arxiv.org/abs/0808.3125 https://arxiv.org/abs/0808.3125) as fluctuations of the DE field into the phantom regime must be controlled or offset assuming one does not want the total energy density to be negative. That turns out to be hard.