You don’t even know what you’re talking about. First of all, the Ethiopian Catholic Church is ORIENTAL, not Eastern - her patrimony is from the ORIENTAL ORTHODOX, not the Eastern Orthodox. .
This is irrelevant to the question at hand because the Holy See’s Annuario Pontificio gives the following list of **Eastern Catholic Churches **with residence and of countries (or other political areas, consisting of more than country) in which they possess an episcopal ecclesiastical jurisdiction (date of union or foundation in parenthesis):
**Alexandrian liturgical tradition
Coptic Catholic Church (patriarchate): Cairo, (163,849), Egypt (1741)
Ethiopian Catholic Church[1] (metropolia): Addis Ababa, (208,093), Ethiopia, Eritrea (1846) **Antiochian (Antiochene or West-Syrian) liturgical tradition
Maronite Church[2] (patriarchate): Bkerke, (3,105,278), Lebanon, Cyprus, Jordan, Israel, Palestinian Authority, Egypt, Syria, Argentina, Brazil, United States, Australia, Canada, Mexico (union re-affirmed 1182)
Syriac Catholic Church[3] (patriarchate): Beirut,(131,692), Lebanon, Iraq, Jordan, Kuwait, Palestinian Authority, Egypt, Sudan, Syria, Turkey, United States and Canada, Venezuela (1781)
Syro-Malankara Catholic Church[4] (major archiepiscopate): Trivandrum, (412,640), India, United States (1930)
Armenian liturgical tradition:
Armenian Catholic Church[5] (patriarchate): Beirut, (375,182), Lebanon, Iran, Iraq, Egypt, Syria, Turkey, Jordan, Palestinian Authority, Ukraine, France, Greece, Latin America, Argentina, Romania, United States, Canada, Eastern Europe (1742)
Chaldean or East Syrian liturgical tradition:
Chaldean Catholic Church[6] (patriarchate): Baghdad, (418,194), Iraq, Iran, Lebanon, Egypt, Syria, Turkey, United States (1692)
Syro-Malabar Church[7] (major archiepiscopate): Ernakulam, (3,902,089), India, Middle East, Europe and America (date disputed)
Byzantine (Constantinopolitan) liturgical tradition:
Albanian Greek Catholic Church (apostolic administration): (3,510), Albania (1628)
Belarusian Greek Catholic Church (no established hierarchy at present): (10,000), Belarus (1596)
Bulgarian Greek Catholic Church[8] (apostolic exarchate): Sofia,(10,107), Bulgaria (1861)
Byzantine Church of the Eparchy of Križevci[9] (an eparchy and an apostolic exarchate): Križevci, Ruski Krstur (21,480) + (22,653), Croatia, Serbia and Montenegro (1611)
Greek Byzantine Catholic Church[10] (two apostolic exarchates): Athens, (2,325), Greece, Turkey (1829)
Hungarian Greek Catholic Church[11] (an eparchy and an apostolic exarchate): Nyiregyháza, (290,000), Hungary (1646)
Italo-Albanian Catholic Church (two eparchies and a territorial abbacy): (63,240), Italy (Never separated)
Macedonian Greek Catholic Church (an apostolic exarchate): Skopje, (11,491), Republic of Macedonia (1918)
Melkite Greek Catholic Church[12] (patriarchate): Damascus, (1,346,635), Syria, Lebanon, Jordan, Israel, Brazil, United States, Canada, Mexico, Iraq, Egypt and Sudan, Kuwait, Australia, Venezuela, Argentina (1726)
Romanian Church United with Rome, Greek-Catholic[13] (major archiepiscopate): Blaj, (776,529) Romania, United States (1697)
Russian Catholic Church[14]: (two apostolic exarchates, at present with no published hierarchs): Russia, China (1905); currently about 20 parishes and communities scattered around the world, including five in Russia itself, answering to bishops of other jurisdictions
Ruthenian Catholic Church[15] (a sui juris metropolia[16], an eparchy[17], and an apostolic exarchate[18]): Uzhhorod, Pittsburgh, (594,465), United States, Ukraine, Czech Republic (1646)
Slovak Greek Catholic Church (metropolia): Prešov, (243,335), Slovak Republic, Canada (1646)
Ukrainian Greek Catholic Church[19] (major archiepiscopate): Kyiv, (4,223,425), Ukraine, Poland, United States, Canada, Great Britain, Australia, Germany and Scandinavia, France, Brazil, Argentina (1595).
So the Ethiopian Catholic Church is in fact, an Eastern Catholic Church.