Understanding the origins of biodiversity has been an aspiration since the days of early naturalists. The immense complexity of ecological, evolutionary, and spatial processes, however, has made this goal elusive to this day. Computer models serve progress in many scientific fields, but in the fields of macroecology and macroevolution, eco-evolutionary models are comparatively less developed. We present a general, spatially explicit, eco-evolutionary engine with a modular implementation that enables the modeling of multiple macroecological and macroevolutionary processes and feedbacks across representative spatiotemporally dynamic landscapes. Modeled processes can include species' abiotic tolerances, biotic interactions, dispersal, speciation, and evolution of ecological traits. Commonly observed biodiversity patterns, such as α, β, and γ diversity, species ranges, ecological traits, and phylogenies, emerge as simulations proceed. As an illustration, we examine alternative hypotheses expected to have shaped the latitudinal diversity gradient (LDG) during the Earth's Cenozoic era. Our exploratory simulations simultaneously produce multiple realistic biodiversity patterns, such as the LDG, current species richness, and range size frequencies, as well as phylogenetic metrics. The model engine is open source and available as an R package, enabling future exploration of various landscapes and biological processes, while outputs can be linked with a variety of empirical biodiversity patterns. This work represents a key toward a numeric, interdisciplinary, and mechanistic understanding of the physical and biological processes that shape Earth's biodiversity.