I'd install the backsplash.

Hood vents can be installed/replaced without affecting the backsplash.

Are the hood and tile going to be around each other? If so, it might be easier to install the hood and then tile around it rather than install the tile and fit the hood into it.

Thank you both. The tile will go beneath the cabinets, then will have to rise to fit just under the hood at 32" above the counter (specs for hood and cooktop), and will continue in a straight line to the doorway. We're putting calcatta pencil trim along the top of the tile in that section. Basically, the hood will "rest on" the backsplash in that section, but the tile isn't "wrapping around" the hood, if that makes sense.

PPF thanks to CFH, with respect to my initial elevation drawing of the wall, that ship sailed a while back. At this point I'm just in survival mode and wanting to get rid of this man and this project.

" and we have our HOA ARC breathing down our necks. "

They have a say on your interior schedule? Who is CFH?

If the hood rests on the top of the tile, I think I can understand why they want to install the hood first. The hood is a more unmoveable and unalterable object. It is easier to adjust the tile a wee bit to the hood than to adjust the hood a wee bit to the tile. It would be a bummer to have the tile installed and need to trim it by a fraction of an inch to get the hood to fit right. Or that tiny extra gap instead of actually resting on the edge of the tile. If they were not supposed to fit together, I may think it would be fine to just install the tile. But they are supposed to fit together.

And just curiousity. You said our tile guy. Does that mean that person is not one of your contractors subs? You are handling that person seperate from your contractor?

1. CFH. maybe contractor from hell?

@beesneeds Thank you. You've made good points, it's just that the hood was supposed to have been completed four months ago, fully custom. It was never done, and now we are already a month past the finish date for our HOA permit for the entire house construction. A few weeks ago, CFH said his carpentry guy never did build it, and now we have to accept a hood from another company because the finish carpentry fellow is sick. We have lost twelve weeks of time with absolutely nobody working on the house at all.

The tile guy is a subcontractor through the CFH, but because the CFH has been so disorganized, I had to contact the tile guy directly to make sure the floors and bathrooms were done correctly. He's really good and I'm scared if I can't get him, CFH will go grab yet another neanderthal from a street corner to do it.

Yes, CFH is Contractor from hell

The math should math, if you have the specifications for everything, including the tile, the grout joint size, and the overall hood size. Most would build the hood slightly tall, so the chimney or shroud could be be scribed to fit. There's no reason the tile cannot be done first, with someone competent using a tape measure and level.