Hi
I am not sure what you mean about the "personal guarantee"?
When someone borrows money, someone else usually guarantees it, not the person taking out a loan. For example, a company's loan would usually be guaranteed by the directors. A child's mortgage, might be guaranteed by their parents.
I am not a lawyer, but as I understand it, your wife is liable for the loan. They can ask her to sell her assets, including her 50% share in the family home, to pay off the loan. It would be legally difficult to do this, especially if there is a mortgage in place.
A better outcome for all would be where you negotiate a solution. Although you are not liable for your wife's debts, you could offer a small amount to settle the debt.
Brendan