'Encountering ETI' weaves together scientific knowledge and spiritual faith in a cosmic context. It explores consequences of Contact between terrestrial intelligent life (TI) and extraterrestrial intelligent life (ETI). Humans will face cosmic displacement if there are other complex, technologically advanced intelligent beings in the universe; our economic structures and religious beliefs might need substantial revision. On Earth or in space, humans could encounter benevolent ETI (solicitous of our striving for maturity as a species) or malevolent ETI (seeking our land and goods to benefit themselves, claiming that their 'superior civilization' gives them the right) - or meet both types of species. Earth Encounters of the Third Kind described by witnesses (including Native American elders) suggest that both may have arrived already: some have been accused of shutting down US and USSR ICBM missiles to promote peace; others of mutilating cattle or abducting people, perhaps to acquire physiological data on biota for scientific study or for other, unknown purposes. Scifi movies such as Avatar and novels like The Martian Chronicles describe humans as malevolent ETI aliens: we do to others what we fear others will do to us. A shared and evolving spiritual materiality could enable humanity to overcome cosmic displacement, and guide TI and ETI in a common quest for meaning and wellbeing on cosmic common ground.