Accurate forecasting of tropical cyclones (TCs) remains challenging due to limited satellite observations probing TC structure and difficulties in resolving cloud properties involved in TC intensification. Recent research has demonstrated the capabilities of machine learning methods for 3D cloud reconstruction from satellite observations. However, existing approaches have been restricted to regions where TCs are uncommon, and are poorly validated for intense storms. We introduce a new framework, based on a pre-training--fine-tuning pipeline, that learns from multiple satellites with global coverage to translate 2D satellite imagery into 3D cloud maps of relevant cloud properties. We apply our model to a custom-built TC dataset to evaluate performance in the most challenging and relevant conditions. We show that we can - for the first time - create global instantaneous 3D cloud maps and accurately reconstruct the 3D structure of intense storms. Our model not only extends available satellite observations but also provides estimates when observations are missing entirely. This is crucial for advancing our understanding of TC intensification and improving forecasts.