This order is enforceable in court by one spouse against the other. If the spouse ordered to pay the debt doesn’t and the other spouse ends up having to pay it, the paying spouse can sue his or her ex to get back the money paid (because the divorce decree ordered the ex to pay the debt).
The order does not however change any liability that the spouses have to the creditor, in this case, the IRS. The IRS does not have to abide by the divorce decree nor does the decree “legally” remove the joint liability. That means that if you and your ex owe taxes jointly, the IRS is free to collect from either party.
The solution then, is to seek reimbursement from your ex. This could be in the form of a cash payment or even in the form of a property exchange. Assuming you and your ex are still on good terms, the two of you should be able to work out an amicable arrangement.
If this is the case, there’s probably no need to involve the courts. If you want the arrangement documented, you can simply draw up a simple agreement on paper for you and your ex to sign.
Many times, however, the ex isn’t quite so agreeable, seeing the situation as a windfall in his or her favor.
Your only recourse in this situation would be to take your ex to court. You can do this either in court via a civil suit or you may also want to consider going back to divorce court and asking the judge find your ex in contempt of court and/or to modify your property settlement to make up for the loss.