You might like to try 'Only You' (1994). It's about a woman who chases a stranger to Italy because she has a weird idea that he is her soul-mate.

The theme of the house servants doing a scam is just like "Fitzwilly" with Dick Van Dyke and Barbara Feldon.
A very different plot, but the setup is the same; and also just a few years later.

I'd Rather Be Rich
Move Over, Darling
That Touch of Mink
Send Me No Flowers
The Seven Year Itch
Teacher's Pet (1958)
Man's Favorite Sport?
The Marriage-Go-Round
Under the Yum Yum Tree
How to Murder Your Wife
A Guide for the Married Man
*** The trouble with reality is there is no background music. ***

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There's also THREE COINS IN THE FOUNTAIN from 1954.
I'm the kind of guy, when I move - watch my smoke. But I'm gonna need some good clothes though.

Seconded!

It was about 10 years later, but there's AVANTI with Jack Lemmon.
I'm the kind of guy, when I move - watch my smoke. But I'm gonna need some good clothes though.

A preliminary list:
American farce in Italy "It Started in Naples" with Clark Gable and Sophia Loren;
especially, "Roman Holiday" with Audrey Hepburn and Gregory Peck
and possibly, "Buona Sera, Mrs. Campbell," with Gina Lollobrigida pursued by 3 American actors.
Italian farce in the U.S. "Houseboat" with Cary Grant and Sophia Loren;
Italian farce in Paris "Love in the Afternoon" with Audrey Hepburn and Gary Cooper
Enjoy !
"A bride without a head !"
"A wolf without a foot !"

"Gidget" ,

With Sandra Dee & Bobby Darin: If A Man Answers, That Funny Feeling,
With Rock Hudson: Pillow Talk, Lover Come Back, Strange Bedfellows
Other good ones: The Thrill Of It All, Gidget