We introduce a new class of self-sustained states, which may exist as single solitons or form multisoliton clusters, in driven passive cylindrical microresonators. Remarkably, such states are stabilized by the radiation they emit, which strongly breaks spatial symmetry and leads to the appearance of long polychromatic conical tails. The latter induce long-range soliton interactions that make possible the formation of clusters, which can be stable if their spatial arrangement is non-collinear with the soliton rotation direction in the microcavity. The clusters are intrinsically two-dimensional and, also, spatially rich. The mechanism behind the formation of the clusters is explained using soliton clustering theory. Our results bring fundamental understanding of a new class of multidimensional cavity solitons and may lead to the development of monolithic multi-soliton sources.