This is a civil case in Texas.
See Texas Rules of Civil Procedure, Rule 193.3(d) and guidance. In the current PDF May 1 2022 those are on pages 199 and 121 respectively and I reproduce them below.
My understanding is that defense counsel (D) could have asserted privilege when Plaintiffs' counsel (P) originally told him about the link. In doing so, D would have identified the inadvertently produced material and the privilege asserted.
Then P would be obliged to promptly return (or delete as it's digital) that material and any copies. Inevitably in this case P would object to the claim of privilege over any material potentially or apparently 'responsive' to discovery (e.g. a folder called "digital copy of defendant's phone" or texts mentioning keywords). The judge would hear from both parties and make a ruling on the material.
In reality, D did not do that.
Also, D made no contemporaneous objections to P's request to admit particular material in court (note P did not request the admission of the entire trove, only particular pieces of it).
Instead, the day after the "Perry Mason moment", D filed an 'emergency motion of protection' and a request for a mistrial. The judge denied both. D asked for another ten days to review the material. The judge said no to ten days but said D could have a D a day to make a start and then they could discuss whether more were needed. She said she would not make a blanket protection order over the entire trove without knowledge of what was in it.
hearing here https://www.youtube.com/watch?v=dKbAmNwbiMk
Incidentally, P claimed there was an earlier inadvertent production of some other material and in relation to that material rule 193.3(d) was followed by both parties. (from 7mins in that clip.)
Texas Rules of Civil Procedure
193.3(d) Privilege not waived by production. A party who produces material or
information without intending to waive a claim of privilege does not waive that
claim under these rules or the Rules of Evidence if - within ten days or a shorter
time ordered by the court, after the producing party actually discovers that such
production was made - the producing party amends the response, identifying the
material or information produced and stating the privilege asserted. If the producing
party thus amends the response to assert a privilege, any party who has obtained
the specific material or information must promptly return the specified material or
information and any copies pending any ruling by the court denying the privilege.
Rule 193.3(d) is a new provision that allows a party to assert a claim of privilege to material or information produced inadvertently without intending to waive the privilege. The provision is commonly used in complex cases to reduce costs and risks in large document Page 122 productions. The focus is on the intent to waive the privilege, not the intent to produce the material or information. A party who fails to diligently screen documents before producing them does not waive a claim of privilege. This rule is thus broader than Tex. R. Evid. 511 and overturns Granada Corp. v. First Court of Appeals, 844 S.W.2d 223 (Tex. 1992), to the extent the two conflict. The ten-day period (which may be shortened by the court) allowed for an amended response does not run from the production of the material or information but from the party’s first awareness of the mistake. To avoid complications at trial, a party may identify prior to trial the documents intended to be offered, thereby triggering the obligation to assert any overlooked privilege under this rule. A trial court may also order this procedure.
This rule imposes no duty to supplement or amend deposition testimony. The only
duty to supplement deposition testimony is provided in Rule 195.6.
Any party can request a hearing in which the court will resolve issues brought up
in objections or withholding statements. The party seeking to avoid discovery has the burden of proving the objection or privilege.