This past April, social media users shared a video that was said to show a homeless man brutally beating his dog in Santa Fe, New Mexico. Within days, the dog’s owner, Chris McMurtry, began receiving threats from animal-lovers. Some commenters on the local Nextdoor platform suggested that someone should offer to buy the corgi mix, named Desi, while others proposed stealing her, in order to save her life. Within weeks, Desi went missing.

Mr. McMurtry immediately phoned the Street Homeless Animal Project, a nonprofit advocate for homeless people and their pets. “The worst has happened,” he said.

As Santa Fe, like many cities across America, grapples with how to address homelessness, tensions are spiking over the welfare of homeless people’s pets. And some residents, frustrated by what they

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