Summary: The study of palaeoclimates relies on information sampled in natural archives such as deep sea cores. Scientific investigations often use such information in multi-stage analyses, typically with an age model being fitted to a core to convert depths into ages at stage one. These age estimates are then used as inputs to develop, calibrate or select climate models in a second stage of analysis. Here, we show that such multi-stage approaches can lead to misleading conclusions, and develop a joint inferential approach for climate reconstruction, model calibration and age estimation. As an illustration, we investigate the glacial-interglacial cycle, fitting both an age model and dynamical climate model to two benthic sediment cores spanning the past 780 kyr. To show the danger of a multi-stage analysis, we sample ages from the posterior distribution, then perform model selection conditional on the sampled age estimates, mimicking standard practice. Doing so repeatedly for different samples leads to model selection conclusions that are substantially different from each other, and from the joint inferential analysis. We conclude that multi-stage analyses are insufficient when dealing with uncertainty, and that to draw sound conclusions the full joint inferential analysis should be performed.