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Python, 222 bytes

(202 characters)

import re
def f(s):y=[w[0]in'ˈˌ'and w or' '+w for w in re.split('([ˈˌ]?[bdfhjklmnprstvwzðŋɡʃʒθ]?[aeiouæɑɔəɛɜɪʊʌː]*)',s)[1::2]];return[[x[i-1:i]or' 'for x in y]for i in range(max(len(w)for w in y),0,-1)]

Returns an array of unicode characters with an array for each row (containing single spaces for each space required)

Not sure where one can get decent output online yet (and I haven't even got the tools to test it properly here either).