The A section of ‘Bernie’s Tune’ is in Dm (the harmonic minor of F) while the B section is in F.
Getting ready for improvisation, it’s instructive to compare Mulligan’s take on this song to the original. Miller’s original has much more of a Spanish feel both rhythmically and harmonically. Bars 5 to 8 are particularly “Spanish” because he uses the notes in the “Spanish Phrygian Scale” with it’s distinctive second and third degree.
The chord in bars 3&4 is unusual. Miller sounds an E dominant7 (an altered II chord) that sort of “resolves” the flat five melody note of bars 1&2. Mulligan plays an altered VI chord (Bb) which actually makes more sense harmonically and adds a bit extra punch.
There is a wonderful collection of original Gerry Mulligan manuscripts at the Library of Congress web site (loc.gov)