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The creditor would first have to bring a lawsuit against you. You can defend if any of the charges are wrong or have some other defense. If not, they will get a judgment against you which will automatically become a lien against your house. If the card is in one one of your names but the house is owned by both, the lien is just on record until you sell the house and then it is enforecable only against the one of you from his/her share of the proceeds. If the card is by both and the house as well with a mortgage, they are unlikely to act because the mortgage would have preference and the foreclosure would not bring money for the judgment as a usual result.
The judgment can be enforced against any personal property, joint or not, with the debtor's name on it.
Texas only allows wage garnishment for child support, certain federal taxes and government backed student loans.
This is information only, NOT legal advice. No attorney-client relationship has been created. Consult an attorney in your state for legal advice regarding your issue.
Practical matter: even if you are both on the title and the judgment is against both of you......it would be too costly and time consuming to foreclose on your home. It is easier to attach a bank account of other personal property.
Suggestion: negotiate the account by reducing it for quick payment and/or time payments...if so get the settlement agreement in writing