I see. Thank you for clarifying that, Eric.
There is typically a 6 month waiting period in California before you may obtain a divorce. During this waiting period, your wife may resurface. If she does not, then you can normally set the case for a hearing and if she does not appear, you can seek a default judgment from the court.
An agent acting with POA authority for a principal has a fiduciary obligation to act in the principal's best interests. Since you are an opposing party in this action, it would not typically be appropriate for you to sign any legal documents for your wife as her POA. If you did and the court accepted the documents, the judgment could be set aside by her at a later date and she could pursue damages against you for breaching your fiduciary obligations to her, so you should not typically sign any documents in this case for her.
Instead, if you pursue a default judgment, it should be granted by the court after the waiting period. Any pleadings should be sent to her at her last known address. The most difficult part is often having the opposing party personally served, which it appears you have already achieved. Since she has notice of the suit, if she choses not to oppose it, you should be able to obtain a default judgment in the case.
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