Motivated by [CH23], we provide a construction of the Brownian Web [TW98, FINR04], i.e. a family of coalescing Brownian motions starting from every point in 1R2 simultane-ously, as a random variable taking values in a space of (spatial) 1R-trees. This gives a stronger topology than the classical one (i.e. Hausdorff convergence on closed sets of paths), thus providing us with more continuous functions of the Brownian Web and ruling out a number of potential pathological behaviours. Along the way, we introduce a modification of the topology of spatial 1R-trees in [DLG05, BCK17] which makes it a complete separable metric space and could be of independent interest. We determine some properties of the characterisation of the Brownian Web in this context (e.g. its box-counting dimension) and recover some which were determined in earlier works, such as duality, special points and convergence of the graphical representation of coalescing random walks.