Reformed and Catholic Eucharist...

The Reformed tradition accepts the four councils of Nicea,  Constantinople, Ephesus 1, and Chalcedon.  In his 1559 edition of Institutes of the Christian Religion, John Calvin wrote:

… we willingly embrace and reverence as holy the early councils, such as those of Nicaea, Constantinople, Ephesus I, Chalcedon, and the like, which were concerned with refuting errors-in so far as they relate to the teachings of faith. For they contain nothing but the pure and genuine exposition of Scripture, which the holy fathers applied with spiritual prudence to crush the enemies of religion who had then arisen.

The ecumenical council at Ephesus in 431, regarding the Lord's Supper, stated in part:

Proclaiming the death according to the flesh of the only begotten Son of God, that is Jesus Christ, and professing his return to life from the dead and his ascension into heaven, we offer the unbloody sacrifice in the churches and so proceed to the mystical thanksgivings and are sanctified having partaken of the holy flesh and precious blood of Christ, the saviour of us all. This we receive not as ordinary flesh, heaven forbid, nor as that of a man who has been made holy and joined to the Word by union of honor, or who had a divine indwelling, but as truly the life-giving and real flesh of the Word. For being life by nature as God, when he became one with his own flesh, he made it also to be life-giving, as also he said to us [in John 6:52]: “Amen I say to you, unless you eat the flesh of the Son of man and drink his blood” .

Roman Catholics read transubstantiation and a misunderstanding of “sacrifice” into the Ephesus 431 quote and thereby judge that Reformed Protestant worship is not catholic, i.e. that the Reformed churches reject the council's teaching.  But that is not the case.  Looking to the Church of England’s Holy Communion service as found in the 1662 BCP (still in use) to help clarify the right understanding of this mystery one finds:

WE do not presume to come to this thy Table, O merciful Lord, trusting in our own righteousness, but in thy manifold and great mercies. We are not worthy so much as to gather up the crumbs under thy Table. But thou art the same Lord, whose property is always to have mercy: Grant us therefore, gracious Lord, so to eat the flesh of thy dear Son Jesus Christ, and to drink his blood, that our sinful bodies may be made clean by his body, and our souls washed through his most precious blood, and that we may evermore dwell in him, and he in us. Amen.

THE Body of our Lord Jesus Christ, which was given for thee, preserve thy body and soul unto everlasting life. Take and eat this in remembrance that Christ died for thee, and feed on him in thy heart by faith with thanksgiving.

THE Blood of our Lord Jesus Christ, which was shed for thee, preserve thy body and soul unto everlasting life. Drink this in remembrance that Christ’s Blood was shed for thee, and be thankful.

The Reformed confession of the Church of England further explains the nature of the sacrament: 
 The Thirty-Nine Articles of Religion; Art. 28:

THE Supper of the Lord is not only a sign of the love that Christians ought to have among themselves, one to another, but rather it is a sacrament of our redemption by Christ’s death: insomuch that to such as rightly, worthily, and with faith receive the same, the bread which we break is a partaking of the body of Christ, and likewise the cup of blessing is a partaking of the blood of Christ.
Transubstantiation (or the change of the substance of bread and wine) in the Supper of the Lord, cannot be proved by Holy Writ, but is repugnant to the plain words of Scripture, overthroweth the nature of a Sacrament, and hath given occasion to many superstitions.
The body of Christ is given, taken, and eaten in the Supper, only after an heavenly and spiritual manner. And the mean whereby the body of Christ is received and eaten in the Supper is Faith.

 The Sacrament of the Lord’s Supper was not by Christ’s ordinance reserved, carried, about, lifted up, or worshipped.

  
My reading of the Ephesus Council quote:

We “offer the unbloody sacrifice in the churches…” i.e. the Sacrament of the Supper (which proclaims Christ’s sacrifice on the cross) is offered to the believers in the churches as real spiritual food for the soul’s nourishment. Note the word “unbloody”, i.e. not corporeal blood. In the same way, in preaching the Gospel, the “sacrifice of Christ is offered” for forgiveness of sins to all that hear it and believe.

This we receive not as ordinary flesh, heaven forbid, nor as that of a man who has been made holy and joined to the Word by union of honor, or who had a divine indwelling, but as truly the life-giving and real flesh of the Word.

Note: “not as ordinary flesh”… in other words not actual physical flesh as one ordinarily would understand… “heaven forbid” i.e. put away the thought that this is physical flesh of a real man, even one “joined to the Word by union of honor.”

Jesus explains to his disciples after their confusion and the objections raised by many of the listens to the words of John 6:52 quoted in Ephesus 431:

61 But Jesus knowing in himself that his disciples murmured at this, said unto them, Doth this cause you to stumble?
62 What then if ye should behold the Son of man ascending where he was before?
63 It is the spirit that giveth life; the flesh profiteth nothing: the words that I have spoken unto you are spirit, are are life.

Thus we receive the body and blood of Christ in the Supper by faith “only after an heavenly and spiritual manner.” (39 Articles. 28)

The only sacrifice offered to God by the clergy or the people is expressed in the final prayer of the Holy Communion in the 1662 BCP service:

“O LORD and heavenly Father, we thy humble servants entirely desire thy fatherly goodness mercifully to accept this our sacrifice of praise and thanksgiving; most humbly beseeching thee to grant, that by the merits and death of thy Son Jesus Christ, and through faith in his blood, we and all thy whole Church may obtain remission of our sins, and all other benefits of his passion. And here we offer and present unto thee, O Lord, ourselves, our souls and bodies, to be a reasonable, holy, and lively sacrifice unto thee; humbly beseeching thee, that all we, who are partakers of this holy Communion, may be fulfilled with thy grace and heavenly benediction. And although we be unworthy, through our manifold sins, to offer unto thee any sacrifice, yet we beseech thee to accept this our bounden duty and service; not weighing our merits, but pardoning our offences, through Jesus Christ our Lord; by whom, and with whom, in the unity of the Holy Ghost, all honour and glory be unto thee, O Father Almighty, world without end. Amen.”