Practical examples of ARMA model
ARMA models are useful when you need to model a Signal plus Noise situation where the signal is an AR process and the noise models sensor noise. The overall model is an ARMA model.

See HL Van Trees, Detection, Estimation, and Modulation Theory, vol 4 Array Processing. He gives an example of a Spatial AR process sensed by noisy sensors. The overall model is ARMA

ARMA model are also useful in situations where you have strong nulls n your spectrum like multi path and you have a noise like signal