In the non-line of sight (NLOS) ultraviolet (UV) scattering communication, the received signals exhibit the characteristics of discrete photoelectrons due to the extremely large path loss. We design and demonstrate an NLOS UV scattering communication system in this work, where the receiver-side signal detection is designed based on a discrete-time Poisson channel model. In our system, a laser and multiple photomultiplier tubes are employed as the optical transmitter and detector, respectively. Furthermore, we design algorithms for pulse-counting, synchronization, channel estimation and $LLR$ computation for hardware realization in FPGA board. Simulation results are provided to evaluate the proposed system design and specify the system key parameters. We perform field tests for real-time communication with the transmission range over $1$km, where the system throughput reaches $1$Mbps.