Yes, you can reintroduce another beam splitter at the end and "lose" the path information, and provided you aren't measuring anywhere along the path, you get wa

Yes, you can reintroduce another beam splitter at the end and "lose" the path information, and provided you aren't measuring anywhere along the path, you get wave interference at the end even if you split the beam along two paths in the beginning. Look up the quantum eraser experiment.