I just checked that setting wol to PHY activity works great with my NIC (ethtool -s eth0 wol gp) : if I unplug ethernet cable before shutting down, the box shuts down completely and wakes up as soon as I replug it.
My NIC seems to be the same as yours :
04:00.0 Ethernet controller [0200]: Realtek Semiconductor Co., Ltd.
RTL8111/8168B PCI Express Gigabit Ethernet controller [10ec:8168] (rev 01)
I just checked that setting wol to PHY activity works great with my NIC (ethtool -s eth0 wol gp) : if I unplug ethernet cable before shutting down, the box shuts down completely and wakes up as soon as I replug it.
My NIC seems to be the same as yours :
04:00.0 Ethernet controller [0200]: Realtek Semiconductor Co., Ltd.
RTL8111/8168B PCI Express Gigabit Ethernet controller [10ec:8168] (rev 01)
Weird...