Optical chaos offers a promising approach to establishing secure communication at high data rates in a shared physical channel, like optical fibers and free space. However, the required synchronization between the transmitter and the receiver can be severely impaired by the nonidealities of the optical link. In particular, free-space optical communications are affected by atmospheric turbulence, which causes beam scintillation and results in time-varying fading of the optical intensity at the receiver side. In this work, we investigate experimentally the propagation of a chaotic signal in an indoor optical link with controllable synthetic turbulence, and we show that the degradation of chaos properties caused by the turbulent environment can be fully mitigated in the optical domain using an adaptive multi-aperture receiver. The proposed receiver integrates a two-dimensional array of optical antennas and a programmable optical processor (POP) on a silicon photonic platform. With respect to a conventional single-aperture receiver, the POP-assisted receiver recovers the complex dynamics of the optical chaos, ensuring a high degree of correlation between the transmitted signal and the received signal, even for a high degree of turbulence. Our results demonstrate the possibility of establishing and maintaining reliable, secure communication in a chaos-based crypto-system in a free space optical link of km-range length.