I totally agree with the Atomos suggestion, they make solid gear, but if you're really looking for reliability, you might want to consider SmallHD instead. Just go with SmallHD, honestly, you cant go wrong with their build quality. I'm always a bit paranoid about screens failing during a shoot and those guys just make stuff that lasts. Just be careful with the cables tho. The micro-HDMI on the ZV-E10 is super flimsy so make sure to pick up a decent cage or a cable clamp. You really dont want to ruin your camera port. Also, monitors eat power like crazy, so I'd suggest getting some high-capacity batteries so you arent constantly worrying about it dying on you mid-take. Better safe than sorry...

The new released Atomos Shinobi II is the best choice for ZV-E10.

I'd suggest looking at the Feelworld F6 Plus for the Sony ZV-E10. At under $200 it offers a lot of bang for the buck. The 5.5" touchscreen is reasonably bright at 500 nits and it has all the essential monitoring tools. The HDMI input accepts up to 4K input. Build quality is quite good for the price. It's a very capable budget-friendly monitor for this camera.

For the Sony ZV-E10, a good monitor should complement its features like its flip-out screen, compact size, and versatility for both video and stills. Look for a monitor with good color accuracy, high resolution (at least 1080p), and brightness for outdoor use. Consider models with touch functionality or customizable buttons for easier control. Brands like SmallHD, Atomos, and FeelWorld offer options that cater well to the ZV-E10's needs, providing additional features like focus peaking and waveform monitoring for professional videography.