We report the experimental evidence of the existence of a random strange attractor in a fully developed turbulent swirling ow. By defining a global observable which tracks the asymmetry in the of angular momentum imparted to the ow, we can reconstruct the associated turbulent attractor and then follow its route towards chaos. We further show that the experimental attractor can be modeled by stochastic Duffing equations, that match the quantitative properties of the experimental flow, namely, the number of quasi-stationary states and transition rates among them, the effective dimensions, and the continuity of the Lyapunov exponents. Such properties can be recovered neither using deterministic models nor using stochastic differential equations based on effective potentials obtained by inverting the probability distributions of the experimental global observables. A random map extracted from the experimental time series of the turbulent ow exhibits qualitatively same stochastic bifurcation structure as the experimentally observed transitions. Our findings open the way to low-dimensional dynamical system modeling of systems featuring a large number of degrees of freedom and multiple quasi-stationary states.