Apple's macOS Catalina causing problems with select eGPU setups
Despite overall feature and performance improvements for external GPUs, the new macOS Catalina is now failing to work correctly with certain combinations of Mac and GPU card.

A Sonnet eGPU with the Sapphire 580 in Apple's own developers' kit
Apple has been steadily improving how its Macs work with external graphics processing units, eGPUs, and in general macOS Catalina continues that effort. But, with the release of Catalina, some users are finding that their Macs may now no longer drive their eGPUs as expected.
The issue is to do with specific combinations of Mac and eGPUs. From AppleInsider research, it appears that Radeon 570 and 580-based eGPUs are having problems with the Mac mini most commonly affected.
Typical problems reported by many users include an inability to boot up the Mac with a display connected to the eGPU. If the Mac will start with the monitor attached, it immediately restarts. Others report their Macs freezing.
Our own direct testing can confirm that the problems are happening on a Sonnet eGPU with the Sapphire 570 card that was in Apple's own developer's kit from two years ago.
Through both our own testing and the reports of users, AppleInsider can report that Vega 56, Vega 64, and Vega VII cards in any eGPU enclosure of suitable power aren't impacted by the problem. Additionally, an eGPU installed and connected to a Mac, with the display connected directly to the Mac and not through the GPU will still accelerate workflows like video processing with no issue.
Apple has been clear that not all eGPU cards and enclosures will work with all Macs -- the most notable omission being anything made by Nvidia. However, Apple also publishes a list of compatible systems and that includes some of the devices currently experiencing problems.

One combination of MacBook Pro and eGPU that Apple itself sells
One significant factor in that compatibility list is to do with the power requirements of the eGPUs. However, power does not appear to be an issue here as the same problems with the Polaris-based Radeon 570 and Radeon 580 are reproducible in the Sonnet's 350W and 650W variants.
Currently, it seems most likely that the issue is tied to drivers for Polaris AMD cards, but as yet Apple has not commented.

A Sonnet eGPU with the Sapphire 580 in Apple's own developers' kit
Apple has been steadily improving how its Macs work with external graphics processing units, eGPUs, and in general macOS Catalina continues that effort. But, with the release of Catalina, some users are finding that their Macs may now no longer drive their eGPUs as expected.
The issue is to do with specific combinations of Mac and eGPUs. From AppleInsider research, it appears that Radeon 570 and 580-based eGPUs are having problems with the Mac mini most commonly affected.
Typical problems reported by many users include an inability to boot up the Mac with a display connected to the eGPU. If the Mac will start with the monitor attached, it immediately restarts. Others report their Macs freezing.
Our own direct testing can confirm that the problems are happening on a Sonnet eGPU with the Sapphire 570 card that was in Apple's own developer's kit from two years ago.
Through both our own testing and the reports of users, AppleInsider can report that Vega 56, Vega 64, and Vega VII cards in any eGPU enclosure of suitable power aren't impacted by the problem. Additionally, an eGPU installed and connected to a Mac, with the display connected directly to the Mac and not through the GPU will still accelerate workflows like video processing with no issue.
Apple has been clear that not all eGPU cards and enclosures will work with all Macs -- the most notable omission being anything made by Nvidia. However, Apple also publishes a list of compatible systems and that includes some of the devices currently experiencing problems.

One combination of MacBook Pro and eGPU that Apple itself sells
One significant factor in that compatibility list is to do with the power requirements of the eGPUs. However, power does not appear to be an issue here as the same problems with the Polaris-based Radeon 570 and Radeon 580 are reproducible in the Sonnet's 350W and 650W variants.
Currently, it seems most likely that the issue is tied to drivers for Polaris AMD cards, but as yet Apple has not commented.
Comments
It is time Timmy & Co get off their high horses and make an affordable Mac that will accept standard cards and standard storage, including user upgradable memory.
It would also really help if the started talking with Nvidia again.
... and indeed, it all seems like a PITA. It seems that these little boxes are still not the right choice for heavy GPU usage. If you want a full-size computer from Apple, you now get to fork over twice the cost of the already too expensive 2013 Mac Pro. What’s that? If we can’t afford a workhorse Apple computer, that computer isn’t for us? Fine. Show me another Mac that is appropriate. Needs an eGPU... which is currently hackish and unreliable, because Apple seem not to be testing OS features before releasing their latest OS to the public... again.
This is progress at Apple? If they’re going to go this route (using an eGPU add-on as a justification for their anorexic machines being the only semi-affordable offerings), they really must get this eGPU stuff to be simple and reliable. It needs to just work.
Apple has had a glaring disinterest in gaming, and now they want people to pay for a game subscription service. Wait. What? Is there anything in that subscription that even remotely compares to PC gaming? I assume not (?), but I haven’t looked (I have ZERO interest in subscription services). Heavy GPU gaming demands more than what Apple’s anorexic machines can offer, and these eGPUs don’t seem to be a simple solution. I can only assume the games on offer will never be up to par with the offerings on PC. Is that a sustainable model for a gaming subscription?
Razr makes great eGPU enclosures AND laptops known for crazy fast desktop like internals. This is not anorexic macs, it is laptops in general.
I am excited about getting a TB3 equipped apple laptop, possibly even a 13” pro or low end 15” pro, instead of my usual cranked top end 15”. A Razr enclosure and a thumping Sapphire card will demolish the Vega20 for the same price as buying the 15” with Vega 20. Rivalling the new MacPro non dual chip GPU options for $$$$ of dollars cheaper.
The most exciting part is i can buy a new graphics card in 2-3yrs, or heck whenever the next big jump drops for 20% or less of the price of a new laptop. Let’s be clear, this will mean stomping the GPU of other’s MacPro setups for 5-10% of the cost.
This means a much faster return on investment for video editors, animators, graphics designers etc. paid for in one small job. Final Cut Pro just got Metal support and will use the eGPU.
As for NVidia, they knowingly screwed Apple with bad chips. I had one of the Macbook Pros with said GPU. Apple will not be going back and they do not have to, their own GPU is making waves now and AMD is coming into their own finally.
Apple Arcade - I am running it on my 4k Apple TV with A10x (at 1080p) and you know what, the games look and move great. My PS4 has better games no doubt, FFXV and HZD are visually way better than Oceanhorn 2 but that is more to do with style and studio budget i suspect. Apple drops an A14x in next year at the 1yr update to Apple Arcade... its going to be edgy stuff. TBH the best part of Apple Arcade is that i can use my PS4 controller, that more than any amount of GPU is the game changer, that has changed the game.
Bit Monster. 11 Bit. They could pick the carcass of Chair. Camoflauge. Beam Dog. Red Hook, Rayark. There are a dozen other small devs that if funded properly and framed with good producers could pump out a library of fantastic cross-platform iOS/macOS/tvOS titles that could make the Apple TV and Apple gaming in general a *great* option.
Instead they make six TV shows, half of them cringey and end up giving it all away because no one is going to rent that platform when the Disney+ TV service already obviously won.
The irony is they would have had the truly "play anywhere" goal of the Switch covered.
The percentage of Mac users that use eGPUs is no doubt TINY. If I was relying on one, I'd sure as hell wait a while before upgrading my OS knowing I'm in a minority, at least until I'm 100% sure everything is compatible. Anything else would be irresponsible, and my OWN fault.
Just because there are some bugs going on with Catalina and certain hardware doesn't mean this is an overall problem with eGPUs on the Mac in general.