I was told by a Marine OR that I could not have an application out with the Navy and Marines at the same time, words...
That's because a few recruiters will just not help you if they don't think you are a promising candidate for selection and by telling them that you are still considering other avenues is the opposite of that.
To the OP, when it comes to Officer accession, recruiters are the gate-keepers.
Since I applied to multiple branches and I have a few theories, my keys to success were:
a) Make sure the timing is right. Know the times between submitting the application, finding out about your acceptance and having to sign the dotted line.
b) Knowing that you are medically qualified in the branch you are interested in. (Ex. different vision standards in USAF and Navy). I recommend getting a full physical from a doctor and a full "eye" physical from an optometrist (for aviation).
c) Understand what MEPS is. MEPS is like a clearing house for applicants looking to enter the armed services. They serve the military, but are not owned by any of the branches in particular. If you go to MEPS for the AF, you don't have to visit again for the Navy, however, you will have to potentially sign paperwork that moves your MEPS files from the AF folder to the Navy one.
d) Your personal statements for the MC, Navy and AF are all different. Leadership should still be a common theme. Navy and AF will be the most similar, with the MC putting a strong emphasis on
leading Marines.
e) Make sure your letters of recommendation properly reflect the branch you are applying to.