Hi Brendan McDonnell,
I've done the more testing with IAR and LPCXpresso IDE and let me demonstrate the testing result.
The demo code is periph_ethernet demo which is from the LPCOpen library, as I can test with different IDE and I'm not family with the Mbed OS 5.7.5. And the Eva board is LPC4088 OEM board and debug tool is LPC-Link 2.
In both of LPCXpresso and IAR IDE, the debug behavior will be lost when click restart button after executing the power down the PHY: LAN8720, as the Fig 1and Fig 2 show.

Fig 1

Fig 2
To recover the debug behavior, I should repower the board and external debugger, then breakpoint feature will become available, in another word, I've not encountered the issue like yours.
About the new issue you talked about, I can replicate the issue you describle, the debugger will lost its behavior during initializing the Ehternet module and PHY chip.

Fig 3
To recomever it, clicking terminate all debug sessions button to stop the current debug behavior, then start a new round of debugg.
Have a great day,
TIC
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