Paper available: P.J.W. Hofstra and J. van Oosten, Ordered PCA's and Realizability Toposes http://www.math.uu.nl/people/jvoosten/papers.html Abstract: The concept of Ordered PCA (Partial Combinatory Algebra) is defined; it is a generalization of ordinary PCA's. The construction of Realizability Toposes for OPCA's is straightforward. Two 2-categories OPCA and OPCA+ are defined, OPCA+ being a lluf subcategory of OPCA. Both have a 2-monad I on them. It is shown that the category of realizability triposes over opca's with Set-indexed exact functors is equivalent to the Kleisli category of I on OPCA, whereas the category of realizabiloity triposes with geometric morphisms is the Kleisli category for I on OPCA+. This extends and analyzes results in the theses of Pitts and Longley. As an application we obtain an elegant tripos presentation of Menni's chain of toposes, constructed as exact completions.