Let $X$ be a compact Kähler manifold and $\Delta$ be a $ℝ$-divisor with simple normal crossing support and coefficients between $1/2$ and $1$. Assuming that $K_X+\Delta$ is ample, we prove the existence and uniqueness of a negatively curved Kahler-Einstein metric on $X\setminus \mathrm{Supp}\left(\Delta \right)$ having mixed Poincaré and cone singularities according to the coefficients of $\Delta$. As an application we prove a vanishing theorem for certain holomorphic tensor fields attached to the pair $(X,\Delta )$.