6I25 --recover

There doesn't seem to be a linux version of mesa flash in the 5i25 firmware archive from the mesanet 6i25 page.

github.com/LinuxCNC/mesaflash

Is the latest but the one from the ISO should be OK

It could possibly be failed hardware of BIOS related

What does

lspci -v

show?

Well that brings up some strangeness of this install
It doesn't have lspci.
a search in Synaptic package manager shows no results for it.

Do you know what library it might be hiding in?

Thank you for that Tommy!

Here are the results:
Without the 6i25 card installed:
00:00.0 Host bridge: Intel Corporation 8th Gen Core 4-core Desktop Processor Host Bridge/DRAM Registers [Coffee Lake S] (rev 08)
00:02.0 VGA compatible controller: Intel Corporation 8th Gen Core Processor Gaussian Mixture Model
00:14.0 USB controller: Intel Corporation 200 Series/Z370 Chipset Family USB 3.0 xHCI Controller
00:16.0 Communication controller: Intel Corporation 200 Series PCH CSME HECI
00:17.0 SATA controller: Intel Corporation 200 Series PCH SATA controller [AHCI mode]
00:1c.0 PCI bridge: Intel Corporation 200 Series PCH PCI Express Root Port (rev f0)
00:1d.0 PCI bridge: Intel Corporation 200 Series PCH PCI Express Root Port (rev f0)
00:1d.3 PCI bridge: Intel Corporation 200 Series PCH PCI Express Root Port (rev f0)
00:1f.0 ISA bridge: Intel Corporation Device a2ca
00:1f.2 Memory controller: Intel Corporation 200 Series/Z370 Chipset Family Power Management Controller
00:1f.4 SMBus: Intel Corporation 200 Series/Z370 Chipset Family SMBus Controller
03:00.0 Ethernet controller: Realtek Semiconductor Co., Ltd. RTL8111/8168/8411 PCI Express Gigabit Ethernet Controller (rev 15)


With the 6i25 installed:
00:00.0 Host bridge: Intel Corporation 8th Gen Core 4-core Desktop Processor Host Bridge/DRAM Registers [Coffee Lake S] (rev 08)
00:01.0 PCI bridge: Intel Corporation Skylake PCIe Controller (x16) (rev 08)
00:02.0 VGA compatible controller: Intel Corporation 8th Gen Core Processor Gaussian Mixture Model
00:14.0 USB controller: Intel Corporation 200 Series/Z370 Chipset Family USB 3.0 xHCI Controller
00:16.0 Communication controller: Intel Corporation 200 Series PCH CSME HECI
00:17.0 SATA controller: Intel Corporation 200 Series PCH SATA controller [AHCI mode]
00:1c.0 PCI bridge: Intel Corporation 200 Series PCH PCI Express Root Port (rev f0)
00:1d.0 PCI bridge: Intel Corporation 200 Series PCH PCI Express Root Port (rev f0)
00:1d.3 PCI bridge: Intel Corporation 200 Series PCH PCI Express Root Port (rev f0)
00:1f.0 ISA bridge: Intel Corporation Device a2ca
00:1f.2 Memory controller: Intel Corporation 200 Series/Z370 Chipset Family Power Management Controller
00:1f.4 SMBus: Intel Corporation 200 Series/Z370 Chipset Family SMBus Controller
01:00.0 PCI bridge: Texas Instruments XIO2001 PCI Express-to-PCI Bridge
02:00.0 DPIO module: Device 2718:5125 (rev 01)
05:00.0 Ethernet controller: Realtek Semiconductor Co., Ltd. RTL8111/8168/8411 PCI Express Gigabit Ethernet Controller (rev 15)

I still am not understanding why the ethernet controller stops working with the card installed.

The only reason for ethernet to stop working when inserting any add on card is if the assigned IRQ is the same and one of the devices does not play nice.
Solutions:
-Insert the add on card into another slot, if possible
-In BIOS disable automatic assignment of IRQ, then try changing the IRQ for the add on card, boot and check if ethernet is working, if not choose a different IRQ. Not much choice there, but do avoid 13, it is used for some stuff that will halt the system when the flag is set.