Careful how you answer...
What happens when an insured sues their insurance company for a bad faith denial of coverage? Usually the company will give a reason why they denied coverage. But what if that reason is that they denied coverage based on conversations and legal advice from their attorneys? Does a plaintiff have a right, through discovery, to inspect those conversations? Two forces are at play here: the sanctity of the attorney-client relationship, and the right for civil litigants to have access to broad discovery.
Hawley Insurance Company v. Contravest
In Hawley Insurance Company, the supreme court answered just that question. To make a long story short (but hopefully still accurate): The plaintiff sued the insurance company (in federal court) for a bad faith denial of coverage. The company denied that they acted in bad faith, but rather, they actually acted in good faith by relying on their attorneys advice. The plaintiff filed multiple motions to have the insurance company's files related to these claims, but the company only provided redacted copies because of the attorney-client privilege. The district court granted the motion to compel the files, and the company filed a writ of mandamus with the Fourth Circuit to vacate the order. Because this case is in federal court, because of diversity, South Carolina law applies. Thus the Fourth Circuit asked the SC Supreme Court its opinion on this matter.
The supreme court first explains the need for discovery plus the general rule of attorney client privilege:

They also held that generally the person claiming the privilege must make a prima facie showing that the privilege exists. Then the burden shifts to the other party who has to prove that either the privilege has been waived or an exception applies:

The court then adopts a new framework for the attorney client privilege based on the Arizona Supreme Court:

The party who is claiming that the company acted in bad faith must first make a prima facie showing of bad faith:

The answer to the complaint
Based on the Lee/Hawley test, it appears that how an insured answers the "answer" can affect how this test applies.
The supreme court specifically held that a mere denial of bad faith does not waive the privilege. "We agree, for we find little authority for the untenable proposition that the mere denial of liability in a pleading constitutes a waiver of the attorney-client privilege." Hawley.
However, the Arizona court explained what type of answer might trigger this waiver:

***This is not legal advice. This is not legal advice. This is not legal advice***
Conclusion
The general rule for attorney client privilege in tort cases against insurance companies for bad faith denial of coverage:
The insured must first make a prima facie showing that the insurance company acted in bad faith. (I believe that this "showing" could be done in either the original complaint or an answer).
Then, the insurance company must make a prima facie showing that the privilege actually exists.
If they do show that the privilege exists, then the insured must prove that either a waiver or exception applies.
The specific rule can be found in this case (Hawley) or the Arizona case which the supreme court has adopted (Lee).