A new multireference perturbation approach has been developed for the recently proposed AP1roG scheme, a computationally facile parametrization of an antisymmetric product of nonorthogonal geminals. This perturbation theory of second-order closely follows the biorthogonal treatment from multiconfiguration perturbation theory as introduced by Surjan et al., but makes use of the additional feature of AP1roG that the expansion coefficients within the space of closed-shell determinants are essentially correct already, which further increases the predictive power of the method. Building upon the ability of AP1roG to model static correlation, the perturbation correction accounts for dynamical electron correlation, leading to absolute energies close to full configuration interaction results. Potential surfaces for multiple bond dissociation in H2O and N-2 are predicted with high accuracy up to bond breaking. The computational cost of the method is the same as that of conventional single-reference MP2.