Modes are based on major keys ( and minor, but that's not the issue here).
Just as every major key has its own key signature, which essentially tells which will be the diatonic notes for the piece, there is a relative minor which contains the same notes, thus with the same key signature. Move to notes from not the Aeolian but harmonic or melodic minors, and accidentals become necessary.
All the modes of a certain parent key will, by definition, contain those same notes as the parent key. Thus, it's important to use that same key signature for any mode from it.
Eb Phrygian has its parent key as Cb. An awful key to write or read, and to my mind, totally unnecessary. So, if the piece is in Eb Phrygian, the key signature will be that of Cb major. Any notes which do not match those diatonics from it will obviously need accidentals, but the whole point is that at the beginning of the piece there will be a statement "these notes are those which generally will be those used hereinafter." Obviously every note is a flattened one, including C itself (being Cb but landing on a white key on piano).
Yes, Eb Phrygian is a minor key - every Phrygian is a minor key - but just because it's based on Eb does not mean it's anything to do with Eb minor, which contains a different pool of notes. Base everything on the parent major, and it's (slightly) simpler.
ggcg sums it up here - 'oy'... Begs the question - why that mode (from that parent key!)?