Assume an operating system maps user-level threads to the kernel using the many-to-many model where the mapping is done through the use of LWPs. Furthermore, the system allows program developers to create real-time threads. Is it necessary to bind a real-time thread to an LWP?

Yes, otherwise a user thread may have to compete for an available LWP prior to being actually scheduled. By binding the user thread to an LWP, there is no latency while waiting for an available LWP; the real-time user thread can be scheduled immediately.