It was quite large - maybe 2cm. Plump and black, with a fuzzy body, huge eyes and a curious pattern on its wings. We couldn't tell if the wings had patches of white or patches of see through.
Finally a few days later we found a picture of the tiger bee fly Xenox tigrinus . Emma astutely pointed out that tigers have stripes not spots and why wasn't it called a leopard bee fly? Apparently they lay their eggs in the nests of carpenter bees and their hatching larvae then eat the bee larvae. The white stuff on the window screen wasn't eggs, then.