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This work is the first step towards a description of the Gromov boundary ofthe free factor graph of a free product, with applications to subgroupclassification for outer automorphisms. We extend the theory of algebraiclaminations dual to trees, as developed by Coulbois, Hilion, Lustig andReynolds, to the context of free products; this also gives us an opportunity togive a unified account of this theory. We first show that any $\mathbb{R}$-treewith dense orbits in the boundary of the corresponding outer space can bereconstructed as a quotient of the boundary of the group by its duallamination. We then describe the dual lamination in terms of a band complex oncompact $\mathbb{R}$-trees (generalizing Coulbois-Hilion-Lustig's compactheart), and we analyze this band complex using versions of the Rips machine andof the Rauzy-Veech induction. An important output of the theory is that theabove map from the boundary of the group to the $\mathbb{R}$-tree is 2-to-1almost everywhere.  A key point for our intended application is a unique duality result forarational trees. It says that if two trees have a leaf in common in their duallaminations, and if one of the trees is arational and relatively free, thenthey are equivariantly homeomorphic.  This statement is an analogue of a result in the free group saying that iftwo trees are dual to a common current and one of the trees is free arational,then the two trees are equivariantly homeomorphic. However, we notice that inthe setting of free products, the continuity of the pairing between trees andcurrents fails. For this reason, in all this paper, we work with laminationsrather than with currents.

Algebraic laminations for free products and arational trees