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Starting from one totipotent cell, complex multicellular organisms form through a series of differentiation and morphogenetic events, culminating in a multitude of cell types arranged in a functional and intricate spatial pattern. To do so, cells coordinate with each other, resulting in dynamics which follow a precise developmental trajectory, constraining the space of possible embryo-to-embryo variation. Using recent single-cell sequencing data of early ascidian embryos, we study this natural variation at the level of a complete interconnected embryo. After developing a robust and biophysically motivated approach to classify cells into distinct transcriptomic states or cell types, a statistical analysis reveals correlations within embryos demonstrating the presence of collective variation. From these intra-embryo correlations, we infer minimal networks of cell-cell interactions using statistical physics models, and from these identify collective modes of gene expression.