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We derive a quantitative rigidity estimate for a multi-well problem innonlinear elasticity with dislocations. Precisely, we show that the$L^{1^{*}}$-distance of a possibly incompatible strain field $\beta$ from asingle well is controlled in terms of the $L^{1^{*}}$-distance from a finiteset of wells, of ${\rm curl}\beta$, and of ${\rm div}\beta$. As a consequence,we derive a strain-gradient plasticity model as $\Gamma$-limit of a nonlinearfinite dislocation model, containing a singular perturbation term accountingfor the divergence of the strain field. This can also be seen as ageneralization of the result of (Alicandro et al. 2018) to the case ofincompatible vector fields.

Geometric rigidity for incompatible fields in the multi-well case and an application to strain-gradient plasticity