An eye for the ladies

You're a detective and you have an assignment to find a client's wife. This good-time gal has found herself a nicer body and is masquerading as some other fellow's wife. So how can you find her? Simple. You get into one male body after another and become a different girl's husband each night. And you're determined to find her even if you have to be every woman's husband to do it! A dull assignment? If you think so, read—
By DARIUS JOHN GRANGER
He was a plump fellow in about the approximate dimensions of a penguin, and as stiffly dressed. Since I'd been an insurance investigator last week but had become a private detective this, and since he was my first potential case, I was needless to say interested.
It's my wife, he said.
Your wife, I repeated, searching for but not finding some of the sharp P.I. dialogue I'd read in the books by Chandler, Evans, Marlowe and others.
You see, we're tourists from another planet. My name is Xlptl.
I just sat there.
Mrs. Xlptl is missing.
Ah, I said, leaning forward. This was something I could understand. Maybe I had heard him wrong about that name.
Missing how? I asked.
Mrs. Xlptl, said my potential client, failed to re-transmigrate.
To do which?