Paul McCartney & Steven Tyler run into each other in the Big Apple … no, it’s not the beginning of a hilarious joke — it happened for real this weekend!

TMZ has obtained video of the two music legends bumping into each other — seemingly by chance — outside the Surrey Hotel on New York City’s Upper East Side Saturday.

Check out the clip … McCartney and his crew are walking toward the entrance — when he seems to notice Tyler standing over to the right.

PM — ever the gentleman — makes his way to ST for a chat … a meeting of the music minds on a shadowy corner of a city street.

Eventually, the two rock legends part ways, and Paul enters the hotel … so not a predetermined meetup here, but a happy accident!

Of course, Paul and Steven are two of the biggest rockstars in the world, boasting 23 Grammys and more than 100 nominations between them — though admittedly, most of those are Paul’s.

Both are members of the Rock and Roll Hall of Fame … and, both have sold millions of records over the years.