All parameters in linear simultaneous equations models can be identified (up to permutation and scale) if the underlying structural shocks are independent and if at most one of them is Gaussian. Unfortunately, existing inference methods that exploit such a non-Gaussian identifying assumption suffer from size distortions when the true shocks are close to Gaussian. To address this weak non-Gaussian problem, we develop a robust semi-parametric inference method that yields valid confidence intervals for the structural parameters of interest regardless of the distance to Gaussianity. We treat the densities of the structural shocks non-parametrically and construct identification robust tests based on the efficient score function. The approach is shown to be applicable for a broad class of linear simultaneous equations models in cross-sectional and panel data settings. A simulation study and an empirical study for production function estimation highlight the practical relevance of the methodology.