Well, there are Eastern Church Benedictines and I’m an EC Benedictine Oblate.
The Biritual Benedictine monastery at Chevetogne is highly regarded by the Orthodox and the ROC in particular - an ROC delegation brought in relics of St Seraphim of Sarov for their common veneration. The Russian Orthodox press commented VERY favourably about this Benedictine monastery and its ecumenical work etc. I don’t see any EC priest or bishop getting these kind of accolades from the ROC - ecumenism at its best.
There are EC Carmelites and Archbishop Raya (+memory eternal!) once told me about them as they were in his Archeparchy when he was in the Holy Land.
In fact, the original monastics of Mt Carmel were Greek Orthodox monks (prior to the schism, of course). The Latin monastics who inherited their legacy went to England where they were pilloried by the true Latin religious Orders for being “Eastern rite invaders” in RC jurisdiction.
There is absolutely nothing wrong with Br. David being in the Carmelite Order. While it is a Latin Order, its roots are Eastern and Carmelite scholars have been studying their Order’s Eastern background more closely, including the original Carmelite Rite, which is the Rite of Jerusalem and the monastic practices of the Thebaid as maintained by the early Greek monks (and before them, there were Jewish hermits in the tradition of St Elias).
And the Orthodox Church venerates the icon of Our Lady of Mt Carmel as a miraculous icon too - especially in Horodyschenske in western Ukraine. Russian Catholics, who cannot ever be seen as “Latinized,” have a great veneration for Our Lady of Fatima and the Carmelite Scapular.
Alex