I recently read about the Digitally project, an overlay payment scheme for use on mobile phones, whose goals are to extend mobile phone payments to areas with poor connectivity and reduce transaction fees.
In summary: DigiTally enables two people to do a payment transaction by simply copying short strings of numbers between their phones. This doesn't need either smartphones or a network connection. Either phone can upload the transaction later, once it gets a network connection.
I see the benefit of such a system working in underdeveloped economics, wherein branchless mobile banking is a way of life. Alice and Bob can engage in an offline transaction which is later verified and added to the blockchain once either of them get internet access.
Are there any technical implementations on Bitcoin or any other cryptocurrency which allow for asynchronous transactions such as the above? If not, then how can we imagine it would look like?
Is there a solution to the double spend problem that works on a delay tolerant network?