Bug#618300: Looks like this may be a well-known issue.

[John Logan <John.Logan@nextissuemedia.com>]

Earlier, I failed to find some information relevant to my problem.

The following link describes my symptoms to a tee, but with the RTL8102EL
device.

https://bugs.launchpad.net/ubuntu/+source/linux/+bug/240470

Here's one more specific to the RTL8111:

http://aplawrence.com/Linux/rtl811.html

In any case, it looks like the answer is to:

1.) Download the latest rtl8168/rtl8111 source from Realtek.
2.) Boot into 2.6.32-5-amd64.
3.) Build the driver module.
4.) Remove the r8169.ko module from /lib/modules/2.6.32-5-amd64.
5.) Run depmod.
6.) Run update-initramfs -u -k 2.6.32-5-amd64.
7.) Reboot into 2.6.32-5-xen-amd64 and repeat steps 3-6.
8.) Reboot back into Xen, network's up and running.

So far, no issues with the updated config.