When a pungent 40-foot whale carcass washed up a few beaches away, Stefanie Worwag and Mario Rivera did what any good neighbors would do. They invited the blubbery bottom feeder into the neighborhood and put him up in their backyard.

Then Dr. Worwag cut him open.

Dr. Worwag, a veterinarian, said she and her husband volunteered to host the dead whale on their Port Hadlock, Wash., property amid a spike in West Coast whale strandings that has researchers asking where they can put all the beached beasts.

“It’s a heck of a learning experience,” said Mr. Rivera, a retired police officer. The couple has been watching the whale deflate ever so slightly each day since it was towed to their stretch of shoreline earlier this month.

The quandary of what to do with whale corpses has troubled coastal populations for decades, and has inspired some novel methods for disposing of the behemoths. States have tried burying them on beaches, dumping them in landfills, sinking them at sea, and on one notable occasion that was caught on camera, blowing them up with dynamite, which sent rancid chunks of whale raining down on spectators.