kitchen tiled backsplash before the faucet?
The order of installing something like a kitchen backsplash and the kitchen sink faucet would really depend upon a number of factors and preferences.


Working clearance. If the faucet location is very near to the backsplash it could be difficult to install the tiles with the faucet in place.
Installing tiles has use of mortar, grout and possibly tile adhesive. It would often be desirable to keep gritty and sticky materials like that, and related cleanup, away from shiny sinks and faucets so installation before either of those would be advisable.
In remodel jobs, especially DIY, it can be necessary to get the basic operation of a kitchen back in order before some parts of the construction and finish work is completed. In this case it may be a preference to have a working sink before a backsplash.
Contractor availability. If having work done using contractors it can sometimes be problematic to line up the contractor work schedules to get tile work and plumbing work done in the ideal order in time. So you make compromises to get the work done in the most acceptable span of time.


Point number 4) above reminds me of the numerous times I've seen street work done where all the nice new curbing is all installed and the concrete nicely cured. Then you see another contractor out there cutting a slot though the curb to install the likes of cable drops to the houses, water or sewer line installation or underground mains AC power line feed installation.