Key message Modelling additive genotype-by-environment interaction is best achieved with the use of factor analytic models. With numerous environments and for outcrossing plant species, computation is facilitated using reduced animal models. The development of efficient plant breeding strategies requires a knowledge of the magnitude and structure of genotype-by-environment interaction. This information can be obtained from appropriate linear mixed model analyses of phenotypic data from multi-environment trials. The use of factor analytic models for genotype-by-environment effects is known to provide a reliable, parsimonious and holistic approach for obtaining estimates of genetic correlations between all pairs of trials. When breeding for outcrossing species the focus is on estimating additive genetic correlations and effects which is achieved by including pedigree information in the analysis. The use of factor analytic models in this setting may be computationally prohibitive when the number of environments is moderate to large. In this paper, we present an approach that uses an approximate reduced animal model to overcome the computational issues associated with factor analytic models for additive genotype-by-environment effects. The approach is illustrated using a Pinus radiata breeding dataset involving 77 trials, located in environments across New Zealand and south eastern Australia, and with pedigree information on 315,581 trees. Using this approach we demonstrate the existence of substantial additive genotype-by-environment interaction for the trait of stem diameter measured at breast height. This finding has potentially significant implications for both breeding and deployment strategies. Although our approach has been developed for forest tree breeding programmes, it is directly applicable for other outcrossing plant species, including sugarcane, maize and numerous horticultural crops.