What condition the wall is appearing - sound, bad, localized deterioration? Type of brick? A photo or two will help.

Not sure about shotcrete over clay-based brick. Brick, especially old brick, is soft and will expand with moisture.
Concrete based shotcrete will shrink and is much harder....fear it would crack up on you.

Another option is to install horizontal soil nails, or screw anchors or ties - and then set the head of these anchors out in front of the brick.
Put on the face of the brick a closed cell foam board and then cast a new concrete "waler wall" across the front of the brick and maybe over the top as well.
This say the concrete serves as a "new" wall face and top and also as a waler for the ties - to create a structural element membrane across the original brick wall face to keep it stable.

A full replacement would be the ideal situation however I want to offer some cheaper options as well.

As large cracks have occurred, and the wall has shifted. A replacement maybe is in the order. However, you might want to see if localized repairs make sense to save a few dollars.

The cracks have divided the wall into 3 regions - left/right wing walls and the middle curved wall. The cracks could be initiated by hoop tension, and worsened by soil pressure on each segment, which has strength reduced due to lose in continuity horizontally. You shall determine the tilt of the wall, and see where of the wall are out of plumb, and the lengths. Then you might be able to just take out the worst segments, and replace with the same. You might need to add a few pilasters to stiffen the wall, or use soil nail to hold the wall (it might not looks good though).

Is there a ground water, or runoff problem? No matter what you do, drainage problem must be corrected prior to reconstruction.

Is this a brick wall or a brick-faced wall? It looks like brick facing to me, in which case you can remove the brick. There are several options to rehabilitate it.

I was looking at the wall in Google, if you're going to replace it, why not use a short precast block wall and grade up to the fence? The big tree at the left end should be removed.

I would definitely consider the horizontal soil nails installed through the wall. Technically, they would not be horizontal. You batter them typically around 15 to 20 degrees so the grout doesn't run out. Not sure what "soil screws" are. The deal breaker with this option is whether or not the nails will be getting installed into the neighbor's property or the client's. If the neighbor is on the high-side, then I guess the client will have to work out some sort of easement deal with the neighbor. Also, you will need to bring on a Geotechnical Engineer.

Agree with JAE regarding offsetting the plate for the soil nails off the face of the brick a slight amount. We normally go with 4" for the offset. Then you have one of two options, cast the wall or shotcrete the wall. The formwork for casting would be expensive but if you are clever you could actually come up with a nifty way to couple on to the end of the soil nail bar and extend it out to use as a form-tie and reduce/eliminate the bracing required. Tape the coupled portion of bar before the pour. Then after stripping the forms, you remove the coupled part of the bar and patch the hole. Otherwise you shotcerte it all (behind the plates too) and make sure you specify exactly how you want it finished. Steel trowel finish, rubber float finish, sculpted, stained? All would increase cost but would add to the aesthetic quality. You do not want to leave it as a natural as-shot finish. That would be offensive. No matter which option you go with, put control joints every 15ft or so. They can score the joints in with the shotcrete wall. Add fibers to the mix no matter what to help control the restrained shrinkage effects.

Quote (JAE)

Brick, especially old brick, is soft and will expand with moisture.

So do cohesive soils, possibly even worse but we shoot shotcrete on soil all the time.

Quote (JAE)

Concrete based shotcrete will shrink and is much harder....fear it would crack up on you.

He should definitely add control joints in new concrete/shotcrete wall no matter what. I would add fibers as well.

There is much more sliding across the interface between shotcrete and soils. With brick - the shotcrete will essentially have a mechanical bond tight to the brick so any differential movement will result in stress...and possibly cracking as a result - a much different condition.

Control joints and fiber may be a good idea.

In theory I agree with you. In practice, the slopes that we shoot shotcrete on are many times uneven and undulating with small and large voids along the length. Plenty of room for mechanical interlock.

From a cost stand point I'm not convinced shotcrete and soil nails is the way to go. Typically shotcrete needs high volume to be cost effective. Brick demo & shotcrete is probably about $200/SF (without the nails) because of the small quantity; higher if you want a skate park or sculpted finish. When I worked in CT in the late 80's/early 90's shotcrete was ConnDOT's preferred concrete repair method; don't know if that's still true today.

If you go shotcrete check the contractor's references. Some agencies require American Shotcrete Association (ASA) nozzleman certification.