Picking through overhanging palm leaves and tiptoeing over mangrove roots, the sound of gushing water rumbles into earshot, drowning out the chirp of tropical birds and insects. The rainforest clears, revealing a crystalline rock pool nourished by the white wisps of Pinaisara Falls, Okinawa’s tallest single-drop waterfall. Iriomote, a 110-square-mile island at the southern tip of Japan—closer to Taiwan than Tokyo—is cut off from the mainland by 650 miles of ocean, and doubly fortified by a hilly jungle and maze of rivers. But this paradise has its pitfalls. A visitor…