We propose a novel identification strategy of imposing sign restrictions directly on the impulse responses of a large set of variables in a Bayesian factor-augmented vector autoregression. We conceptualize and formalize conditions under which every additional sign restriction imposed can be qualified as either relevant or irrelevant for structural identification up to a limiting case of point identification. Deriving exact conditions we establish that, (i) in a two dimensional factor model only two out of potentially infinite sign restrictions are relevant and (ii) in contrast, in cases of higher dimension every additional sign restriction can be relevant improving structural identification. The latter result can render our approach a blessing in high dimensions. In an empirical application for the US economy we identify monetary policy shocks imposing conventional wisdom and find modest real effects avoiding various unreasonable responses specifically present and pronounced combining standard recursive identification with FAVARs.