If your ex spouse is deducting the payments as alimony and you are not reporting it as income; you will both hear from the IRS and the IRS will end up making that determination.
I'm of the opinion (I'm not a lawyer) that it is alimony unless the decree requires you to specifically use the alimony money to pay the mortgage. The fact that you may not have sufficient other funds to meet the payment, does not make it not alimony. That is, the provision allowing you to live there, if you make the payments is not specifically tied to the alimony payment.
