There is really only ONE real (in Trek terms) solution to this, and interestingly, the solution was postualted in TNG.
The TNG episode, "Parallels" provides the solution - the 'Mirror Universe' is simply nothing more that a fairly deep quantum reality. And whether the TNG people consciously did it or not, it provides the answers to several of the nagging problems with Mirror, Mirror.
1. Whenever Worf passed from QR to QR, he gets dizzy/almost passes out. I believe in MM, they mention being slightly dizzy as well.
2. With each transition, Worf is in a reality that is increasingly divergent from his core reality:
A birthday party that doesn't exist
Yellow/chocolate cake
The different paintings, etc
Eventually the changes are MAJOR, as Worf goes further down the split realities:
The different console, being married to Troi, Geordi dying, and DR Ogawa.
And the last transition is the biggest of all - A ship that looks quite a bit different; Riker as Captain, Wesley as Weapons Officer, A Cardassian navigator, and
Worf is wearing a different uniform!
That makes Kirk and the landing party materializing in the MU uniforms totally logical now. They didn't beam into a different Universe, they 'simply' beamed into a different quantum reality of the Universe, with their uniforms conforming to that QR. If Spock had been in the party, he just might have found himself with a beard, or Sulu with a scar!
Too bad in Parallels they fell back on the Borg cliche, when after the fissure was busted wide open, and endless Enterprises were entering that Quantum reality, instead of the Borg devastated ship trying to stop Worf returning, an
ISS Enterprise tried to do it, and then about a hundred other ships firing on THEM to stop them from doing so.
