Indoor optical wireless communications have been widely studied to provide high-speed connections to users, where the use of repetition-coded (RC) multiple transmitters has been proposed to improve both the system robustness and capacity. To exploit the benefits of the RC system, the multiple signals received after transmission need to be precisely synchronized, which is challenging in high-speed wireless communications. To overcome this limit, we propose and demonstrate a recurrent neural network (RNN)-based symbol decision scheme to enable a delay-tolerant RC indoor optical wireless communication system. The experiments show that the proposed RNN can improve the bit-error-rate by about one order of magnitude, and the improvement is larger for longer delays. The results also show that the RNN outperforms previously studied fully connected neural network schemes.