An 11th century manuscript, Minuscule 72, contains a marginal note which says (in Greek): "Because, in the Gospel, according to a report of Diodore and Tatian and various other holy fathers, this is added: 'And another, having taken a spear, pierced his side, and there came out water and blood.' Chrysostom also says this."
It has been suggested that the scribe who wrote (or copied?) this comment mistook the words 'dia 4' (διά δʹ = Diatessaron; note that Greek uses letters to designate numbers) for the name 'Diodorus' (Διόδωρου, Diodorou). While the scholion traces this variant to Tatian, we have no evidence for the insertion of anything corresponding to John 19:34 at this point. On the contrary, the evidence we have of the Diatessaron shows the presence of the words after Jesus death. In Ephrem's commentary it can be localized somewhere between 27:53 and 27:58, while in the Arabic version it comes after 27:54. But even if the Diatessaron could be established as a source, it is still a puzzle why these diverse witnesses inserted the words here.
In narrative context, it could be argued that the placement of the variant is a bit awkward grammatically and narratively.
But the others said, “Wait, let us see whether Elijah will come to save him.”
(οἱ δὲ λοιποὶ ἔλεγον, Ἄφες ἴδωμεν εἰ ἔρχεται Ἠλίας σώσων αὐτόν.)
**But* another, having taken a spear, pierced his side, and there came out water and blood.*
(ἄλλος δὲ λαβὼν λόγχην ἔνυξεν αὐτοῦ τὴν πλευράν, καὶ ἐξῆλθεν ὕδωρ καὶ αἷμα.)
But Jesus crying out again with a great voice, yielded up his spirit.
(ὁ δὲ Ἰησοῦς πάλιν κράξας φωνῇ μεγάλῃ ἀφῆκεν τὸ πνεῦμα.)
Without the piercing sentence everything flows smoothly: "'Wait, let us see whether Elijah will come to save him,' - but Jesus crying out again with a loud voice..." If the variant is included, the last de (δὲ, 'and', 'but') is especially awkward. The reading also presents a jarring contradiction to what was just described: while the bystanders were waiting to see if Elijah would come and save Jesus, someone - in complete opposition to this sentiment - kills Him with his spear (!) We have at least two choices: either this is a very unskillful secondary insertion or it is original and has been eliminated to improve style and remove a difficulty. (Assuming that the variant is original, there is in fact a suggestion that the emphasis on eyewitness testimony in John 19:35 was meant with regard to the time of the piercing, to object to other accounts - like Matthew - who placed it before Jesus' death.) There is no convincing explanation for a secondary addition of this kind.
Finally, we should note that this variant was apparently still known in the West, at least up to the 14th century: Pope Clement V's declaraton of John's gospel as the one which preserved the proper chronological order at the Council of Vienne (1311-1312) is often taken as a condemnation of the textual variant.
Adhering firmly to the foundation of the catholic faith, other than which, as the Apostle testifies, no one can lay, we openly profess with holy mother church that the only begotten Son of God, subsisting eternally together with the Father in everything in which God the Father exists, assumed in time in the womb of a virgin the parts of our nature united together, from which he himself true God became true man: namely the human, passible body and the intellectual or rational soul truly of itself and essentially informing the body. And that in this assumed nature the Word of God willed for the salvation of all not only to be nailed to the cross and to die on it, but also, having already breathed forth his spirit, permitted his side to be pierced by a lance, so that from the outflowing water and blood there might be formed the one, immaculate and holy virginal mother church, the bride of Christ, as from the side of the first man in his sleep Eve was fashioned as his wife, in this way, to the determinate figure of the first and old Adam, who according to the Apostle is a type of the one who was to come, the truth might correspond in our last Adam, that is to say in Christ. This, we say, is the truth, fortified by the witness of that huge eagle which the prophet Ezechiel saw flying over the other gospel animals, namely blessed John the apostle and evangelist, who relating the event and order of this sacrament, said in his gospel: *But when they came to Jesus and saw that he was already dead, they did not break his legs, but one of the soldiers pierced his side with a spear, and at once there came out blood and water. He who saw it has borne witness - his testimony is true, and he knows that he tells the truth -- that you also may believe.*
We, therefore, directing our apostolic attention, to which alone it belongs to define these things, to such splendid testimony and to the common opinion of the holy fathers and doctors, declare with the approval of the sacred council that the said apostle and evangelist, John, observed the right order of events in saying that when Christ was already dead one of the soldiers opened his side with a spear.