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We present ANISE, a method that reconstructs a 3D~shape from partial observations (images or sparse point clouds) using a part-aware neural implicit shape representation. The shape is formulated as an assembly of neural implicit functions, each representing a different part instance. In contrast to previous approaches, the prediction of this representation proceeds in a coarse-to-fine manner. Our model first reconstructs a structural arrangement of the shape in the form of geometric transformations of its part instances. Conditioned on them, the model predicts part latent codes encoding their surface geometry. Reconstructions can be obtained in two ways: (i) by directly decoding the part latent codes to part implicit functions, then combining them into the final shape; or (ii) by using part latents to retrieve similar part instances in a part database and assembling them in a single shape. We demonstrate that, when performing reconstruction by decoding part representations into implicit functions, our method achieves state-of-the-art part-aware reconstruction results from both images and sparse point clouds.When reconstructing shapes by assembling parts retrieved from a dataset, our approach significantly outperforms traditional shape retrieval methods even when significantly restricting the database size. We present our results in well-known sparse point cloud reconstruction and single-view reconstruction benchmarks.