My experience at women's monasteries (Greek* and Serbian**) is that the nave of the monastery church is divided into two parts. The section closest to the iconostasis is for the monastics and the other section farther from the altar is ok for non-monastics and I thought non Orthodox? Come to think of it, I have never been to a monastery in the company of non-Orthodox... I do know that unless we were asked to help read we were expected to stay way in the back. I have a memory of the Gerontissa (Abbess) asking questions and moving people around a little depending on the answer... what she asked and how they responded I cannot say. All to say: it's possible!
* Life Giving Springs, Dunlap, CA (Ephraimite)
** St. Xenia Skete, Wildwood, CA (Serbian)
At St Herman of Alaska (Serbian, Platina, CA) we also stayed in the back but it didn't seem as strict. For strictness, it's hard to beat an Ephraimite monastery!