I saw the current production of The Tempest at The Globe and enjoyed the music that accompanied the show. The inclusion of original incidental music is a tradition that goes way back.
Shakespeare has been fairly well set to music. Verdi did Othello and Falstaff, Romeo and Juliet has been covered by Gounod and Tchaikovsky, Two Gentlemen of Verona shows up in a Schubert song Who is Sylvia (An Sylvia).
Of all the Shakespeare music, Mendelssohn's incidental music for A Midsummer Night's Dream is the most famous.
The traditional wedding recessional music is from the Mendelssohn incidental music. In addition to that warhorse, The Overture and Nocturne are tremendous.
The Overture was written when Mendelssohn was but 17 years old. In my opinion this composition surpasses anything Mozart wrote at a comparable age. Nothing Mozart wrote that early has enjoyed the popularity of Mendelssohn's overture.
Mendelssohn wrote The Overture on his own and completed the rest of the incidental music, under commission from The King of Prussia, 16 years later.
Mendelssohn does a good job of including themes from The Overture but the incidental music is clearly written by a mature master.
I like the song setting of "You Spotted Snakes" but The Nocturne is amongst the best music Mendelssohn ever wrote.
Here are a few links.