In this paper two examples of two independent centered Gaussian processes are given such that at least one of them is not a semimartingale but their sum is a martingale.
"A Gaussian martingale which is the sum of two independent Gaussian non-semimartingales." Electron. Commun. Probab. 20 1 - 5, 2015. https://doi.org/10.1214/ECP.v20-4034