The Chaldean Church, which is in communion with Rome, uses the Anaphora of Addai and Mari.
Yes, it does.

The question of the Anaphora of Ss Addai & Mari has come up regularly, and while I don’t wish to be redundant to other threads and posts, perhaps a few comments might be useful here as a frame of reference.
The original recension of the Anaphora of Ss Addai & Mari, as used by the ACoE, does indeed lack an Institution Narrative, and it was this recension that was the focus of the decision of the Holy Office. The determination was that the Institution is implicit within the Anaphora, and so the Anaphora as a whole unit is considered to be consecratory. The recension of the Anaphora of Ss Addai & Mari used by the Chaldean Church is nearly identical, but does have an Institution Narrative, that was
added by Rome at the time of union in the late 16th century. Within the past few years, the Holy Synod of the Chaldean Church promulgated some liturgical restorations and it is interesting to note that it was decided, after much scholarly research, to retain the Institution Narrative in that Anaphora. So, it’s still there, although its position has been shifted a bit to what is considered a more appropriate place within the Anaphora.
By contrast, in the Roman Canon, the Institution Narrative is quite explicit. The Epiklesis is implicit and comes before the Institution Narrative rather than explicit coming after it (*1). While the other Roman Rite Eucharistic Prayers are perhaps a bit more explicit in wording, it is still positioned before the Institution Narrative and is of a different character than is Epiklesis in the various Eastern and Oriental liturgies. To my knowledge, the Epiklesis in the Roman Mass (*2) seem to have always been placed before the Institution Narrative and therefore has to be considered more implicit than explicit, irrespective of the text. That’s still the case, Novus Ordo and additional EPs notwithstanding. Not one of them is truly explicit in the Eastern/Oriental sense, either in text or placement.
*1: There was also a [thread=535966]thread[/thread] earlier this year in the EC forum concerning the Epiklesis that might be of some interest.
*2: This seems to be true in the Ambrosian and other Western Rites too, although apparently the Mozarabic Rite differs a bit: some say it has no epklesis at all while others say the prayer “post pridie” (after the Institution Narrative) is intended to be an implicit epiklesis.