Complete Kähler metric question for regular loci and complements

Let $X$ be a weakly pseudoconvex Kähler complex space. Its regular locus is denoted by $X_{\operatorname{reg}}$, and an analytic subset of a complex space is denoted by $A$. Complete Kähler metric question. Does there exist an example for which $X_{\operatorname{reg}}$ does not admit a complete Kähler metric? Furthermore, when $X$ is a weakly pseudoconvex Kähler manifold, does there exist an analytic subset $A$ such that $X\setminus A$ does not admit a complete Kähler metric?

The question asks whether the complete-Kähler-metric conclusions known for compact or $1$-convex weakly pseudoconvex spaces, and for complements of analytic subsets in those settings, can fail in the general non-compact case.