Motivated by [G. Cannizzaro, M. Hairer, arXiv:2010.02766], we provide an
alternative characterisation of the Brownian Web (see [T\'oth B., Werner W.,
Probab. Theory Related Fields, '98] and [L. R. G. Fontes, M. Isopi, C. M.
Newman, and K. Ravishankar, Ann. Probab., '04]), i.e. a family of coalescing
Brownian motions starting from every point in $\mathbb R^2$ simultaneously, and
fit it into the wider framework of random (spatial) $\mathbb R$-trees. We
determine some of its properties (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.
Along the way, we introduce a modification of the topology of spatial $\mathbb
R$-trees in [T. Duquesne, J.-F. Le Gall, Probab. Theory Related Fields, '05]
and [M. T. Barlow, D. A. Croydon, T. Kumagai, Ann. Probab. '17] which makes it
Polish and could be of independent interest.