Escaping spaces in tab-completion in a Cygwin shell subprocess on GNU Emacs on Windows

Question

I use GNU Emacs on Windows and Cygwin as my shell. If I create a Cygwin shell subprocess with M-x shell, I get a proper bash prompt. If I type this:

cd ~/My<tab>

it tab-completes to

cd ~/My Documents/

which is correct in Emacs/Tramp, but useless in the Cygwin bash shell subprocess, since the space has to be escaped with a backslash.

How do I get this tab-completion to automatically insert a backslash?