As one word: both [ˈʌɪ̯ɻəʃmn̩]
Yes, they really are homophones — but only when said as compound words rather than as two separate words in succession. As compound words, the American pronunciation of Irishman and Irishmen alike is usually just [ˈʌɪ̯ɻəʃmn̩] in connected speech, no matter whether singular or plural. Only the stressed syllable has a distinctive, non-reduced vowel, as is the normal case in English.
The other two syllables are very short, and the final syllable has been further reduced so that it has only a syllabic consonant [n̩], not a vowel proper.
As two words: [əˌnʌɪ̯ɻəʃ ˈmæə̯n] vs. [səˌmʌɪ̯ɻəʃ ˈmɛn]
It’s the unstressed reduction that makes all the difference. These are pronounced differently than one would pronounce “an Irish man” [əˌnʌɪ̯ɻəʃ ˈmæə̯n]
or “some Irish men” [səˌmʌɪ̯ɻəʃ ˈmɛn].
Only when separate words giving man/men primary stress does it work differently, because that’s the only way for the distinction between [ˈmæə̯n] and [ˈmɛn] to appear. Plus the singular man is a little bit longer because of its stressed [æə̯] diphthong under ae-tensing, something that doesn’t happen in the unstressed part of compound words.
But what about Rosa’s roses?
Peter Shor comments that for him, the singular Irishman has what is in effect the unstressed vowel of Rosa’s, while the plural Irishmen has what is in effect the unstressed vowel of roses.
For many people, [ə] and [ɨ] are alternate phonetic allophones of the same underlying phoneme /ə/. It is a matter of some controversy whether there actually exists a distinct /ɨ/ phoneme instead of just an [ɨ] allophone of /ə/; for those who hear Rosa’s and roses as different words, there may well be. For those who do not, there isn’t.
For more about this, see the highly cited 2007 linguistics paper “Rosa’s roses: reduced vowels in American English” by MIT’s Edward Flemming and CalTech’s Stephanie Johnson.
That same unstressed vowel [ɨ] might also occur in some utterances of Irish, so [ˈʌɪ̯ɻɨʃ]. The degree of vowel reduction in English unstressed syllables is always highly variable, and words said in isolation don’t sound the same as words said in connected speech.
If we’re going to discuss actual sounds, and it seems we must, then we’ll need to use the phonetic notation used by specialists not the phonemic notation used by dictionaries. This can be quite complicated, but it’s the only way to show what we mean in writing. I provide three references to help reading, writing, and listening:
- [READING] The Wikipedia page on the International Phonetic Alphabet.
- [WRITING] A way to type those symbols using the Full IPA Keyboard.
- [LISTENING] The University of Edinburgh’s incredibly useful Sound Comparisons website as mentioned by phoneticist John Wells on his blog. See for example how these words are actually said in different accents including Irish ones, with narrow phonetic transcriptions and actual sound clips provided: