Is it permissible to sell a house, which is not yet completed, in the wife's name to the bank, then the bank sells it to the husband, so that the husband obtains a subsidized mortgage loan from the fund to complete the construction of the house, with the wife participating in paying one-third of the house's value?

It is not permissible to deal with a bank that lends with interest (riba), whether you pay the interest yourself or the fund pays it, as this is forbidden by Islamic law. As for the bank selling the house in installments, there is no objection to that. However, it is not permissible to resort to deception by transferring the ownership of the house to your wife's name so that the bank buys it and gives you the price; this is a deception to circumvent riba and a fraud against the bank and the fund, which makes it one of the major sins.