DHAKA, Bangladesh—A fire engulfed a food and beverage factory outside Bangladesh’s capital, killing at least 52 people, many of whom were trapped inside by an illegally locked door, fire officials said Friday. The blaze began Thursday night at the five-story Hashem Foods Ltd. factory in Rupganj, just outside Dhaka, sending huge clouds of black smoke billowing into the sky. Police initially gave a toll of three dead, but then discovered piles of bodies on Friday afternoon after the fire was extinguished. So far 52 bodies have been recovered, but the top two floors of the factory have yet to be searched, said Debasish Bardhan, deputy director of the Fire Service and Civil Defense. He said the main exit of the factory was locked from the inside and many of those who died were trapped. Many workers jumped from the upper floors of the factory, and at least 26 suffered injuries, …