While paganism, or neopaganism, isn’t necessarily secular or interfaith (the word “pagan” can denote a wide range of groups, from the decidedly polytheistic, pantheistic, animistic, or magickal, to the merely “not Judeo-Christian”) it would appear that pagans are up to some interesting monastic stuff:

• Order of the Horae – Pagan Monasticism FAQ
• Journal of a Pagan Monastic
• Twilight and Fire
• Maetrum of Cybele
• The Order of the Sacred Nemeton

I hope we can share notes.