In a small bowl soften tapioca in warm water to cover 1 hour.
While tapioca is soaking, in a small saucepan cook coconut milk, ginger, and sugar over moderate heat, stirring, until sugar is dissolved.
Remove pan from heat and let mixture stand 10 minutes.
Drain tapioca. Peel bananas and quarter lengthwise.
Cut quarters into 1/2-inch pieces.
Heat coconut milk mixture over low heat and gently stir in tapioca and bananas. Cook pudding, stirring (be careful not to break up bananas), until thickened slightly and bananas are heated through, about 5 minutes, and cool. Divide pudding among 6 ramekins and chill, covered, until cold.
