IT IS POSSIBLE to get mortgage in Romania even as a non-resident. One of the banks who provide these financial means is Volksbank, and they are probably the best since their competition doesn't cover 70% of the property value.
If you chose to become resident, then almost all the Romanian banks offer these services. However, becoming a resident implies a few taxes you have to pay to the Romanian state, so you should think about it and talk with a Romanian financial consultant before you jump into that train.
Regards,
Diana Marinescu