We define a natural, purely geometrical bijection between the set solutions of Bethe ansatz equations for the Gaudin magnet chain and the set of arc diagrams of Frenkel-Kirillov-Varchenko. The former set is in natural bijection with monodromy-free sl_2-opers (aka projective structures) on the projective line with the prescribed type of regular singularities at prescribed real marked points (according to Feigin and Frenkel), while the latter indexes the canonical base in a tensor product of U_q(sl_2)-modules (via the Schechtman-Varchenko isomorphism). Both sets carry a natural action of the cactus group, i.e., the fundamental group of the real Deligne-Mumford space of stable rational curves with marked points (by monodromy of solutions to Bethe ansatz equations on the former and by crystal commuters on the latter). We prove that our bijection is compatible with this cactus group action. This is joint work with Nikita Markarian.