Depending on capacity or the extent of the power given to your dad a trust may be possible. Contact a local estate planning attorney.
Probate would be necessary to effectuate the close of the sale or distribute the proceeds
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Contact an attorney right away--the house would require probate if it's not in a trust. If your grandmother can understand and approve the trust, she can sign it, or if the power of attorney allows your father to set up a trust for her benefit, he can have it prepared and it's also important to be sure the house gets transferred to the trust (this can be done with the power of attorney) as well, before her death.
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You may want it to sell after she dies, for a number of reasons not considered in your question (step-up in basis, reduction of capital gains taxes, etc.) So I am not sure this is a case where you want to rush without getting advice after a thorough analysis. It may also be that she may have incurred a large Medi-Cal bill, in which case the value of the house (which would have been excluded for Medi-Cal qualification purposes) would nevertheless be subject to cost recovery if it is in her estate at her death. But in simplest terms, yes, dying while owning a "fee interest" (outright ownership) in real estate of any significant value (more than $40,000 I think) will trigger a probate. Deceased people cannot own property; hence, a Court must be involved in order to complete the legally required retitling. Moving it to a trust may "avoid probate" (a concept which is often oversold) but may create other problems (lack of oversight, challenges of other heirs and beneficiaries, etc.), and selling it while she is alive could create yet more problems (e.g., capital gains taxes).
This is legal information only and not meant to provide legal advice. Many issues that seem straightforward at first are often complicated by facts not revealed in a hypothetical posed by a member of the public. You should always consult directly with your attorney in order to ensure the issue is thoroughly discussed and that the proper course of action is taken.