There is a paper
by Tokunaga that advocates string theory on supermanifolds. Bosonic string theory may be defined on supermanifolds whose super-dimension is (26+k/k), he or she argues. That of course cancels the central charge - since the spin 0 worldsheet fermions carry c=-1 - and she or he also shows modular invariance. However I find it likely that these theories won't be fully physical in spacetime - for example they will violate spin-statistics relations, won't they?
Also, I wonder what the insiders among the readers think, after a year or so, about the status of mirror symmetry of supermanifolds. Has it passed the test of time?