That just confirms my experiences. I've had issues with onboard Realtek cards and linux. On one desktop the Realtek card would work until the box was restarted. You would have to hard power it off and back on for it to work again. That same box worked fine with Windows.
The old RTL8139 which is Realtek's 10/100 card have been rock solid for many years, and the RTL8169S gigabit cards are rock solid too, with a heatsink.
You can't beat Realtek if you're on a budget.