A while ago I asked this question in Math Stackexchange. Since I didn't receive an answer so far, I thought I'd ask it here.

Suppose $Y$ is a proper length space, where every pair of points $x,y\in Y$ can be joined by a *unique* length minimizing geodesic (i.e. global geodesic). Can there still be more than one *local* geodesic joining some two points, or is it necessarily so that the space is also uniquely locally geodesic?

If a uniquely geodesic space which is not uniquely locally geodesic exists, can one even cook up Riemannian manifold as an example?