A length space is a metric space $X$, where the distance between two points is the infimum of the lengths of curves joining them. The length of a curve $c: [0,1] \rightarrow X$ is the sup of $$ d(c(0), c(t_1)) + d(c(t_1), d(t_2)) + \cdots + d(c(t_{N-1}), c(1)) $$ over all $0 < t_1 < t_2\cdots < t_{N-1} < 1$ and $N > 0$.

A geodesic space is a length space, where for each $x,y \in X$, there is a curve $c$ connecting $x$ to $y$ whose length is equal to $d(x,y)$.

A Riemannian manifold $M$ and its metric completion $\overline{M}$ are length spaces. If the Riemannian manifold is complete, then it is a geodesic space.

But is $\overline{M}$ necessarily a geodesic space? If not, what is a counterexample?

This was motivated by my flawed answer to Minimizing geodesics in incomplete Riemannian manifolds

Also, note that if $\overline{M}$ is locally compact, then it is a geodesic space by the usual proof. One example of $M$, where $\overline{M}$ is not locally compact is the universal cover of the punctured flat plane. However, this is still a geodesic space.