  We show that non-stationary Gromov-Witten invariants of $\mathbb{P}^1$ can be
extracted from open periods of the Eynard-Orantin topological recursion
correlators $\omega_{g,n}$ whose Laurent series expansion at $\infty$ compute
the stationary invariants. To do so, we overcome the technical difficulties to
global loop equations for the spectral $x(z) = z + 1/z$ and $y(z) = \ln z$ from
the local loop equations satisfied by the $\omega_{g,n}$, and check these
global loop equations are equivalent to the Virasoro constraints that are known
to govern the full Gromov-Witten theory of $\mathbb{P}^1$.
