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The space $ \mathfrak{ft}_n = \mathbb C^n/\mathbb C $ of $n$ points on theline modulo translation has a natural compactification $\overline{\mathfrak{ft}}_n $ as a matroid Schubert variety. In this space,pairwise distances between points can be infinite; we call such a configurationof points a ``flower curve'', since we imagine multiple components joined intoa flower. Within $ \mathfrak{ft}_n $, we have the space $ F_n = \mathbb C^n\setminus \Delta / \mathbb C $ of $ n$ distinct points. We introduce a naturalcompatification $ \overline F_n $ along with a map $ \overline F_n \rightarrow\overline{\mathfrak{ft}}_n $, whose fibres are products of genus 0Deligne-Mumford spaces. We show that both $\overline{\mathfrak{ft}}_n$ and$\overline F_n$ are special fibers of $1$-parameter families whose genericfibers are, respectively, Losev-Manin and Deligne-Mumford moduli spaces ofstable genus $0$ curves with $n+2$ marked points.  We find combinatorial models for the real loci $\overline{\mathfrak{ft}}_n(\mathbb R) $ and $ \overline F_n(\mathbb R) $. Usingthese models, we prove that these spaces are aspherical and that theirequivariant fundamental groups are the virtual symmetric group and the virtualcactus groups, respectively. The deformation of $\overline F_n(\mathbb{R})$ toa real locus of the Deligne-Mumford space gives rise to a natural homomorphismfrom the affine cactus group to the virtual cactus group.

The moduli space of cactus flower curves and the virtual cactus group