Midwest rhymes with:
addwest, addwest, bequest, eastern-west, farwest, key_west, natwest, northwest, norwest, penwest, quest, request, southwest, stateswest, telequest, telewest, west
Midwest sounds like:
matched, matchett, matsuda, matsushita, mediaset, medicaid, medicate, medicated, mendacity, metagogued, methodist, mideast, midget, midgett, midgette, midsized, midst, mindset, mitigate, mitigated, modesitt, modest, modesta, modestia, modesto, modesty, modisette, monteagudo, mudgett