We construct actions of the spheromorphism group of Neretin (containing Thompson's group V) on towers of moduli spaces of genus zero real stable curves. The latter consist of inductive limits of spaces which are the real parts of the Grothendieck–Knudsen compactification of the moduli spaces of punctured Riemann spheres. By a result of M. Davis, T. Januszkiewicz and R. Scott, these spaces are aspherical cubical complexes whose fundamental groups, the ‘pure quasi-braid groups’, can be viewed as analogues of the Artin pure braid groups. By lifting the actions of the Thompson and Neretin groups to the universal covers of the towers, we obtain extensions of both groups by an infinite pure quasi-braid group, and construct an ‘Euler class’ for the Neretin group. We justify this terminology by constructing a corresponding cocycle.