I used to think a Poisson process is the only point process with constant event rate.

Question

It's not hard to show that the exponential distribution is the only inter-arrival distribution with a constant event rate. And if you consider the distribution of the number of events falling into an interval, you get a Poisson distribution, with the overall process being a Poisson process. So, it followed that the Poisson point process should be the only point process with a constant event rate. But then I thought about a simple compound Poisson point process. To keep things simple, assume that every arrival of the Poisson process, five events happen instead of just one event. Let's call this a deterministically compounded Poisson process. It seems that this process should also have a constant event rate. The expected number of events in any interval should still be independent of any other interval. And if this deterministically compounded point process has a constant event rate, then other kinds of compounding, where the number of events happening at every arrival is some random variable independent of the underlying Poisson process should also have a constant event rate. Can we say then that a Poisson process along with any compound Poisson process and nothing else should have a constant event rate?