Midwest's rhymes with:
bequests, natwest's, northwest's, norwest's, quests, requests, southwest's, west's, arrests, attests, bequests, best's, breasts, chests, crests, digests, divests, fests, guest's, guests, guests', infests, invests, jests, molests, natwest's, nests, northwest's, norwest's, pests, quests, reinvests, requests, rests, retests, southwest's, suggests, test's, tests, tests', vests, west's
Midwest's sounds like:
matsushita's, medicates, metastasis, metastasize, midgets, midgett's, mindsets, mitigates, mitsotakis, montesdeoca