Good question. Being a native speaker, I just pronounced this word a few times to see how I actually pronounce it.
My pronunciation is as follows: пришéдши-и. The part after ''ш'' is as follows: I start pronouncing the ''и'' sound, then it gets quieter, and then loud again, but does not break in the middle. It is like two waves, so to speak.
Pronounce the following word considering it English: pre-shed-shee-ee. This is practically identical to how I pronounce ''пришедшие.'' The syllable ''shee'' is like in ''sheep.''
If I speak fast (i.e., as usual in everyday communications) and use ''пришедшие'' in a sentence, then all vowels practically get swallowed (i.e., are almost not pronounced at all). For example, I pronounce the expression ''пришедшие солдаты'' like ''прьшедшислдаты,'' very fast and barely distinct enough for a native speaker to recognise what I am saying.
Note that there are various dialects in Russia, and mine is the Moscow one, notorious for fast indistinct pronunciation and preference of ''a'' over ''о.''
In other dialects the word ''пришедшие'' may sound differently. To give you an idea how different Russian dialects are, I point to the Novgorod dialect (''новгородский выговор''), in which the city name ''Новгород'' is pronounced exactly as written, with three distinct ''о,'' whereas I pronounce the same word rather as ''Навгрд.''