There are (were) 3 bands called Eucharist:
1) A melodic Death Metal band from Veddige, Sweden.
2) A hardcore band from Richmond, VA, USA.
3) A black death metal band from Australia.
1) The Swedish Eucharist was founded in Veddige, a little city about 40 km from Gothenburg (Göteborg), Sweden sometime in 1989. They released a demo in 1992. The same year a 7-inch EP called "Greeting Immortality" was released by Obscure Plasma Records. It featured song 1 and 3 from the demo.