mg.metric geometry - Intrinsic metric with no geodesics

There is a very simple example of an intrinsic, complete metric space that is not geodesic (read in Ballmann's "Lectures on Spaces of Nonpositive Curvature": it is the graph on two vertices $x,y$, linked by edges $e_n$ of length $1+1/n$.

Of course it does not answer your question, but it may be possible to improve this example to one that does. Call $X_1$ the graph described above, and define $X_{n+1}$ from $X_n$
as follows: $X_n$ has a vertex $x'$ for each vertex $x$ of $X_n$, plus a vertex $v_e$ for each edge $e$ of $X_n$. For each edge $e=(xy)$ of $X_n$ we define edges $f_e^n$ and $g_e^n$ of $X{n+1}$: $f_e^n$ connects $x'$ to $v_e$ and has length $(1+1/n)$ times the original length of $e$, and $g_e^n$ does the same
but replacing $x'$ by $y'$.

Now it should be possible to construct the desired example by a limiting process. For example, take all vertices along the construction: the distance between any two of this points is constant as long it is defined, so we get a metric space. Its completion might be what you want (but I a not so sure of that after witting these lines).