Inland American Real Estate Trust Inc. is the kind of property company that gives some investors bad dreams. It raised $7.9 billion to buy assets it now can’t or won’t sell. The value of its holdings has plunged 60%. It has cut its dividend by three-quarters.

“It’s a zombie,” says Allan Roth, one of 170,000 shareholders of the nontraded real-estate investment trust. Instead of returning money to shareholders, the REIT is plowing proceeds from the few properties it has sold into new purchases or reducing debt, complains the...