Monday Mutant 8: Pearls of Wisdom (crosslink)

Monday Mutants is a series in which I will attempt to experiment with "mutant" puzzles. These could be existing puzzle types with an unusual change in the rules, hybrids combining elements from multiple puzzle types, or puzzle types neither invented nor popularized by Nikoli.
In this Pearls of Wisdom puzzle, the loop may pass through a cell twice, but only if the loop goes straight through that cell both times, once horizontally and once vertically. If the loop passes through a white pearl twice, then it must obey the normal rules (by turning at one or both of the adjacent cells in the loop) both times. The rules are otherwise unchanged.