See [Forcing for IZF in sheaf toposes](https://www.researchgate.net/publication/228380851_Forcing_for_IZF_in_sheaf_toposes) Also [Toposes from Forcing for Intuitionistic ZF with Atoms.](https://arxiv.org/abs/1702.03399) --- Edit: Maybe more references: [Heyting-valued models for intuitionistic set theory](https://link.springer.com/chapter/10.1007%2FBFb0061825) The book "Intuitionistic logic, model theory and forcing" by Fitting. see also the following web-page where some references are also given: [Constructive Set Theory: Forcing, Large Sets, and Mathematics](https://www.nsf.gov/awardsearch/showAward?AWD_ID=0301162) [Topological forcing semantics with settling](http://www.sciencedirect.com/science/article/pii/S0168007211001369?via%3Dihub).