I read the above question in Bob's post as being the main point.
Well, the answer to that is quite simple: no normal lender would lend unless they could repossess and sell in the event of default. So, the mortgage would have to be set up like that. It is, technically possible to sell a part share in a property (and this is where the distinction between owning it pro indiviso, or in defined proportions could be important) but no mainstream mortgagee is going to be interested in that proposition. If you could find someone to lend on a basis of security over only a share in a property, I would suspect the interest rate would be getting close to that of an unsecured loan.
It is not uncommon in commercial property loans, particularly where there were multiple lenders (but also sometimes including equity participants too) to have a situation similar to this. But the hierarchy of who got paid first was always clearly defined. And senior debt always came first. Then subordinate debt and equity a distant third. And what is worthwhile for a £10m commercial investment may not be cost-effective for a typical residential transaction.
Some other things that occur to me, in no particular order.
Scotland has a Matrimonial Homes Act that could give some protection from eviction, I think. Not my area. I assume England has something similar. And, of course, the parties would need to be in a relationship. [edit: 9fingers may have answered that. But you never know]
What happens if the property is sold before the end of the mortgage. Equal shares of any profit? But it could be the case that the mortgagor party still has only a small proportion of equity. So, if they get an equal share, they have leveraged themselves to a far better position than the equity participant. One way of looking at it is that the equity party has made a gift of the use of their capital – not that different from a loan?
Attitudes to repairs maybe different. If say new windows are required the eagerness to do so may well be different if you have 50% equity in the property against someone who, at that point has only 10% (and is paying a mortgage as well).
It is difficult to see how the rights of the person that is putting in the capital can be protected. If the mortgagor has defaulted they are hardly likely to have assets elsewhere. So you get into the realms of insurance, guarantors or cross collateral.
And of course, none of the above constitutes advice in any way. Merely thoughts.