Scattering-based computed tomography (CT) recovers a heterogeneous volumetric scattering medium using images taken from multiple directions. It is a nonlinear problem. Prior art mainly approached it by explicit physics-based optimization of image-fitting, being slow and difficult to scale. Scale is particularly important when the objects constitute large cloud fields, where volumetric recovery is important for climate studies. Besides speed, imaging and recovery need to be flexible, to efficiently handle variable viewing geometries and resolutions. These can be caused by perturbation in camera poses or fusion of data from different types of observational sensors. There is a need for fast variable imaging projection scattering tomography of clouds (VIP-CT). We develop a learning-based solution, using a deep-neural network (DNN) which trains on a large physics-based labeled volumetric dataset. The DNN parameters are oblivious to the domain scale, hence the DNN can work with arbitrarily large domains. VIP-CT offers much better quality than the state of the art. The inference speed and flexibility of VIP-CT make it effectively real-time in the context of spaceborne observations. The paper is the first to demonstrate CT of a real cloud using empirical data directly in a DNN. VIP-CT may offer a model for a learning-based solution to nonlinear CT problems in other scientific domains. Our code is available online.