Using first-principles calculations we investigate the mutual passivation of shallow donor Si and isovalent N in dilute GaAsN alloys. Instead of the recently proposed pairing of Si and N on adjacent substitutional sites (Si(Ga)-N(As)) [K. M. Yu et al., Nat. Mater. 1, 185 (2002); J. Li et al., Phys. Rev. Lett. 96, 035505 (2006)] we find that N changes the behavior of Si in dilute nitride alloys in a more dramatic way. N and Si combine into a deep-acceptor split interstitial, where Si and N share an As site [(Si-N) (As)], with a significantly lower formation energy than that of the Si(Ga)-N(As) pair in n-type GaAs and dilute GaAsN alloys. The formation of (Si-N)(As) explains the GaAs band-gap recovery and the appearance of a photoluminescence peak at approximately 0.8 eV. This model can also be extended to Ge-doped GaAsN alloys, and correctly predicts the absence of mutual passivation in the case of column-VI dopants.