  We show global wellposedness for the defocusing cubic nonlinear Schr\"odinger
equation (NLS) in $H^1(\mathbb{R}) + H^{3/2+}(\mathbb{T})$, and for the
defocusing NLS with polynomial nonlinearities in $H^1(\mathbb{R}) +
H^{5/2+}(\mathbb{T})$. This complements local results for the cubic NLS and
global results for the quadratic NLS in this hybrid setting.
