Summary: Weakly coupled regions of 4d EFTs coupled to gravity are particularly suitable to describe the backreaction of BPS fundamental axionic strings, dubbed EFT strings, in a local patch of spacetime around their core. We study the extension of these local solutions to global ones, which implies probing regions of strong coupling and provides an estimate of the EFT string tension therein. We conjecture that for the EFT string charge generators such a global extension is always possible and yields a sub-Planckian tension. We substantiate this claim by analysing global solutions of 4d strings made up from NS5-branes wrapping Calabi-Yau threefold divisors in either type IIA or heterotic string theory. We argue that in this case the global, non-perturbative data of the backreaction can be simply encoded in terms of a GLSM describing the compactification, as we demonstrate in explicit examples.