Let $(X,\|\cdot\|,\le)$ be a normed, ordered vector space over $\mathbb{R}$ and let $X^+=\lbrace x\in X:x\ge0\rbrace$ denote the (positive) cone in $X$. with a metric $d$ induced by the norm $\|\cdot\|$.
Is there an example of a (positive) cone $X^+$ and normed space $X$ such that $(X^+,d)$ complete and the normed space $X,\|\cdot\|$ is not complete?