You've nothing to worry about, Daisy - in fact quite the reverse! Bomareas including multiflora almost all come from high-altitude areas of the South and Central American tropics where they are used to high humidity. Bomarea multiflora stays in leaf (and even in flower) virtually all year in such conditions, and in the north island of New Zealand where it has a warm moist climate all year it's regarded as a troublesome weed, getting out of hand in gardens and strangling other plants. But your garden must be much less humid than that, and the dry summer air should keep it well in check. It will die back and rest probably until very early next spring.
We grow Bomarea salsilla, and find that dies back around the end of June or even earlier, shooting again in the late winter. It's really not happy in hot dry conditions - whereas your multiflora looks as if it's been revelling in the treatment you've given it.
