NVIDIA GTX 680 2Gb Mac flashed is the best basic option:
I recently acquired a used EVGA 02G-P4-2684-KR and flashed it with the Mac ROM. So far it has worked as expected, the only thing unusual is that the boot screen from the HDMI output has a light green instead of the light gray background. Most issues and answers related to flashing the Mac ROM...
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I flash them in a 5.1 using Windows 10 installed from DVD (NOT USB). Instaling Windows 10 from USB will likely "Brick" the Mac Pro as detailed elsewhere.
AMD RX580 4Gb or 8 Gb are higher end solutions, however, check the card heights as most will partially obscure the only x16 PCIe slot 2 needed for 4x4 NVMe PCIe cards. Gigabyte Aorus range have two cards that do not obscure slot 2 at all. Must use a 2 x 6 pin mini PCIe power to 8 pin "Y" cable with RX580 cards.
The Mac Pro 5.1 will need to have the High Sierra firmware before installing Mojave. If older OS installed, get to 10.11 or fresh install 10.11 prior to HS.
Older Mac OSX installers will need networking disconnecting and the Mac clock changed to a date around 6 months after the installer issue date. Failure to do this will drop the Mac into a boot loop that is a real PITA to get out of !
Download current versions of the HS and Mojave installers from Apple with the updated security certificates that avoid this issue.