Let $\rho_p:G_{\mathbb{Q}_p} \to \text{Gl}_n(\mathbb{Q}_p)$ be semistable representation. In local to global Galois representation, it was asked if one can find a geometric global Galois representation $\rho:G_{\mathbb{Q}}\to \text{Gl}_n(\mathbb{Q}_p)$ such that $\rho\vert_{G_{\mathbb{Q}_p}}=\rho_p$. This doesn't work for one-dimensional representations for cardinality reason, as was pointed out in this answer. In the comments to this answer, BCnrd mentions a concrete obstruction ot liftability
"For a number field $K$ and a (continuous) character $\rho:G_K\to \overline{\mathbb{Q}}_p^\times$ ramified at only finitely many places, the image on inertia at the places away from $p$ is finite. So if unramified at all places dividing $p$ then must have finite image (since inertia subgroups of $G^{ab}_K$ at the non-archimedean places topologically generate a finite-index subgroup, due to finiteness of class groups). So that gives a "concrete" obstruction to globalizing."
Is this obstruction effective? More generally, do we have a good idea which "nice" local representations come from "nice" global representation?