While I agree with Bob's answer, (how could I not, it was his experience) I think the best thing you can do is talk to a few banks. A local one, that keeps its mortgages in its own portfolio. Not one that just writes loans and immediately sells them to be repackaged into mortgage pools for investors.
Be upfront with the bank, and give them all the details.
If I were the bank, I'd be happy to offer such a loan, with the condition that you are offering a first lien on your paid-for half. Think about it. Banks give mortgages with 20% all day long, but also under 10% down with PMI. Fluctuations in housing prices mean the bank can easily lose money when houses go 'under water' i.e. the loan is more than the home value, there is no actual water involved. Typically, a bank will loosen their requirement both for income to debt ratios and credit score when a borrower can show such a large down payment.
If partner cannot/does not make a payment, you are at risk, so of course, you should make a payment. This should all be agreed to in advance. If you make any payment or lend them money to make the next payment, they agree that either (a) your equity goes up by the full payment amount, or (b) they owe you the money at X% rate. Whatever your situation, do this all in writing, with a lawyer, who specializes in real estate. You can't have zero risk in such situation, but can minimize potential damage by doing this.