In this paper we study the backward stability of running a backward stable eigenstructure solver on a pencil $$S(\lambda )$$ that is a strong linearization of a rational matrix $$R(\lambda )$$ expressed in the form $$R(\lambda )=D(\lambda )+ C(\lambda I_\ell -A)^{-1}B$$ , where $$D(\lambda )$$ is a polynomial matrix and $$C(\lambda I_\ell -A)^{-1}B$$ is a minimal state-space realization. We consider the family of block Kronecker linearizations of $$R(\lambda )$$ , which have the following structure $$\begin{aligned} S(\lambda ):=\left[ \begin{array}{ccc} M(\lambda ) &{} {\widehat{K}}_2^T C &{} K_2^T(\lambda ) \\ B {\widehat{K}}_1 &{} A- \lambda I_\ell &{} 0\\ K_1(\lambda ) &{} 0 &{} 0 \end{array}\right] , \end{aligned}$$ where the blocks have some specific structures. Backward stable eigenstructure solvers, such as the QZ or the staircase algorithms, applied to $$S(\lambda )$$ will compute the exact eigenstructure of a perturbed pencil $$\widehat{S}(\lambda ):=S(\lambda )+\varDelta _S(\lambda )$$ and the special structure of $$S(\lambda )$$ will be lost, including the zero blocks below the anti-diagonal. In order to link this perturbed pencil with a nearby rational matrix, we construct in this paper a strictly equivalent pencil $$\widetilde{S}(\lambda )=(I-X)\widehat{S}(\lambda )(I-Y)$$ that restores the original structure, and hence is a block Kronecker linearization of a perturbed rational matrix $${{\widetilde{R}}}(\lambda ) = {{\widetilde{D}}}(\lambda )+ {{\widetilde{C}}}(\lambda I_\ell - {{\widetilde{A}}})^{-1} {{\widetilde{B}}}$$ , where $${{\widetilde{D}}}(\lambda )$$ is a polynomial matrix with the same degree as $$D(\lambda )$$ . Moreover, we bound appropriate norms of $${{\widetilde{D}}}(\lambda )- D(\lambda )$$ , $${{\widetilde{C}}} - C$$ , $${{\widetilde{A}}} - A$$ and $${{\widetilde{B}}} - B$$ in terms of an appropriate norm of $$\varDelta _S(\lambda )$$ . These bounds may be, in general, inadmissibly large, but we also introduce a scaling that allows us to make them satisfactorily tiny, by making the matrices appearing in both $$S(\lambda )$$ and $$R(\lambda )$$ have norms bounded by 1. Thus, for this scaled representation, we prove that the staircase and the QZ algorithms compute the exact eigenstructure of a rational matrix $${{\widetilde{R}}}(\lambda )$$ that can be expressed in exactly the same form as $$R(\lambda )$$ with the parameters defining the representation very near to those of $$R(\lambda )$$ . This shows that this approach is backward stable in a structured sense. Several numerical experiments confirm the obtained backward stability results.