Integrated sensing and communication (ISAC) has recently merged as a promising technique to provide sensing services in future wireless networks. In the literature, numerous works have adopted a monostatic radar architecture to realize ISAC, i.e., employing the same base station (BS) to transmit the ISAC signal and receive the echo. Yet, the concurrent information transmission causes severe self-interference (SI) to the radar echo at the BS which cannot be effectively suppressed. To overcome this difficulty, in this paper, we propose a coordinated cellular network-supported multistatic radar architecture to implement ISAC. In particular, among all the coordinated BSs, we select a BS as the multistatic receiver to receive the sensing echo signal, while the other BSs act as the multistatic transmitters to collaborate with each other to facilitate cooperative ISAC. This allows us to spatially separate the ISAC signal transmission and radar echo reception, intrinsically circumventing the problem of SI. To this end, we jointly optimize the transmit and receive beamforming policy to minimize the sensing beam pattern mismatch error subject to both the communication and sensing quality-of-service requirements. The resulting non-convex optimization problem is tackled by a low-complexity alternating optimization-based suboptimal algorithm. Simulation results showed that the proposed scheme outperforms the two baseline schemes adopting conventional designs. Moreover, our results confirm that the proposed architecture is promising in achieving high-quality ISAC.