Should he sell his house to an Islamic finance company to pay off the remainder of a riba-based loan?
Question

                I live in Canada. Five years ago we committed the crime of buying our house through an interest based mortgage from a local bank here. Relying on many fatawa of scholars here (may Allah forgive them!). 

Two years ago Allah guided us to Islam like if we just knew it. Recently, some Islamic (as they claim) financing companies here started to appear. 

The following is an explanation of their way to convert an interest based mortgage to Islamic financing, based on cooperating: the Islamic company deals with a bank that deals with interest to finance their project. This Islamic company says that its relationship with the bank is not a Riba relation, it is a trade relation. 

This company values the house in the local market; an appointed person licensed by the government does this valuation matter. 

The monthly amount I have to pay is based on the rest of the loan I took and the profit which is about 4.9%. 

Some of this monthly amount I pay goes as profit, and the rest is to be discounted from the actual price of the house. The Islamic financing company’s share is the difference between the price of the house now and what is remaining to be paid of the interest-based loan. The company evaluates the price of the house every one, two or five years. Their share of profit changes according to the number of years I choose. This means that the nisbah of profit is fixed, but the share of every one of us gets evaluated every few years. 

What remains of the mortgage (their share of the house) is $178.000, their profit in the first five years is 49038.88$, and I own of the house what equals 18408.08$ of the house which cost is $178.000. 

The relationship between the company and the bank is unclear. It depends in its essence on buying money with money; as they did not buy the house formally. All these transactions are only on paper between the company and the bank. The house actually has nothing to do with these transactions except that it is mentioned on papers between the two sides. We are really confused, we hope you enlighten us.
Praise be to Allah.We ask Allaah to reward you 
    with good for your keenness to seek what is halaal and to rid yourself of 
    the effects of what you did by taking a loan from the riba-based bank. We 
    will answer your question as follows: 
Firstly: consuming riba and 
    helping others with it is haraam and is in fact a major sin. Muslim narrated 
    in his Saheeh that Jaabir (may Allaah be pleased with him) said: The 
    Messenger of Allaah  (peace and blessings of Allaah be upon him) cursed 
    the one who consumes riba and the one who pays it, the one who writes it 
    down and the two who witness it, and he said: they are all the same. 
A loan in which is 
    stipulated that something extra will be paid to the lender, as the banks do, 
    is haraam and there is no difference of scholarly opinion on this point. 
Secondly: You said that you 
    did that as you were unaware that it is haraam. We hope that this means that 
    there is no sin on you because of that. Allaah says in the verses which 
    speak of the prohibition on riba (interpretation of the meaning): 
“So whosoever receives 
    an admonition from his Lord and stops consuming Riba, shall not be punished 
    for the past; his case is for Allaah (to judge)”
[al-Baqarah 2;275]. 
     
It says in Fataawa 
    al-Lajnah al-Daa’imah li’l- Ifta’ (13/352): With regard to the interest 
    that you took before you came to know that it is haraam, then we hope that 
    Allaah will forgive you for that. As for that which you got after you came 
    to know, you have to get rid of it and spend it on charitable causes. End 
    quote. 
Thirdly: You do not have to 
    sell the house to the company mentioned or to any other in order to pay of 
    the loan. If it is stipulated in the loan contract that the lender will get 
    some benefit, even if it is haraam, it does not invalidate the loan, so your 
    purchase of the house is valid, and your ownership of it is valid, and you 
    do not have to sell it in order to expedite paying off what remains of the 
    loan to the bank, especially when the ways of dealing with the company you 
    mention, are not free of ambiguity. At the end of your question, you 
    mentioned that they will not buy the house in real terms; if the sale and 
    purchase are done on paper and possession is not really taken of the house 
    before it is sold for a second time, there is the fear that the company’s 
    dealing may be a trick to commit riba in the name of selling. 
And Allaah knows best.