I seem to have resolved this issue by installing the alternate kernel module for this chip set available from elrepo.
# /sbin/lspci -nn | grep -i net . . . 01:00.0 Ethernet controller [0200]: Realtek Semiconductor Co., Ltd. RTL8111/8168B PCI Express Gigabit Ethernet controller [10ec:8168] (rev 03) . . .
# /sbin/lspci -n | grep '01:00.0' 01:00.0 0200: 10ec:8168 (rev 03)
Using the the device id (10ec:8168) to search:
http://elrepo.org/tiki/DeviceIDs
shows that this matches:
r8168.ko
pci 10EC:8168 kmod-r8168
# yum whatprovides kmod-r8168 . . .
kmod-r8168-8.031.00-1.el6.elrepo.x86_64 : r8168 kernel module(s) Repo : elrepo Matched from: . . .
# yum install kmod-r8168 . . . Setting up Install Process Resolving Dependencies --> Running transaction check ---> Package kmod-r8168.x86_64 0:8.031.00-1.el6.elrepo will be installed --> Finished Dependency Resolution
Dependencies Resolved
================================================================================ Package Arch Version Repository Size ================================================================================ Installing: kmod-r8168 x86_64 8.031.00-1.el6.elrepo elrepo 73 k
Transaction Summary ================================================================================ Install 1 Package(s)
Total size: 73 k Installed size: 533 k Is this ok [y/N]: y Downloading Packages: . . .
This seems to have cleared up the problem on both hosts but, only time will tell.