The Anaphora of Pope Sixtus and Anaphora of Mor Dionysius all have the essential elements of a traditional Anaphora. These elements include: the Preface prayers to the father, an Institution narrative and an Epiclesis. What is, however different is the fact that the Institution narrative is in the third person and does not contain the words this is my body or this is my blood.
From the Anaphora of Mor Dionysius Bar Salibi (that is used in the Syro-Malankara Church very regularly):
When He prepared for the redemptive passion, he took bread and blessed + + and sanctified + and broke, and called it His Holy Body for eternal life for those who receive it.
And also the cup blended of wine and water, He blessed + + and sanctified + and completed as His Precious Blood of eternal life for those who receive it.
It is important here to note, that while the words this is my body aren't said explicitly. The Priest himself is still acting as the mediator between God and Man (in Latin theological terms, the priest himself is doing what Christ did. That is, he is acting in the person of Christ). Nevertheless, my job here is not to defend the validity of these Anaphoras but rather, raise awareness of them.
In the Syriac-East we accept the Anaphora is valid because they have been passed down from our fathers, brethren, priests and teachers whom were all given authority, from the Apostles themselves, to bind on earth what it is in heaven (Rome has approved the validity of some of these Anaphorae anyway, but approval from Her is not necessary). The important thing here is that the Priest follows the proper traditions and rubrics for celebrating the Liturgy of the Divine Mysteries in the way our great Syriac fathers intended, the rest is a mystery.
It is important to note here, that there is an Anaphora in the East Syriac tradition which lacks an Institution Narrative entirely, that being the Anaphora of Marai Addai & Mari. The Assyrian Church of the East maintains this Anaphora in its original, while the Chaldean Church has added an Institution Narrative to mirror those in the other two Anaphorae used in the East Syriac tradition. Also, a number of scholars suggest that the uniquely Maronite anaphora called Peter III (sometimes called "Sharar" after its first word), which is also of the East Syriac model, may have originally lacked an Institution Narrative as well. Whether this is true or not isn't clear, but what is clear is that Peter III as it has survived has a very unique form of the Institution: it's written in the 2nd Person Singular ("You took bread ..." etc).
For the remaining Anaphorae (those of the West Syriac style), in actual practice the Maronite Church has substituted a standardized text of the Institution Narrative for the original, proper text. This has also been the practice in Syriac CC although in recent years some Anaphorae have had the proper texts restored. Among the West Syriac Churches in union with Rome, it's only the Syro-Malankara that has continuously maintained the use of the actual text proper to each Anaphora.
Through the prayers of the Birth-Giver of God Maryam and Mor Ephrem. God bless you all.