This is an interesting area that's been explored by only a few players (Roland von Straaten and Howard Levy come to mind), but they used standard Richter harps. There's a lot of scope for optimising the expressive feel of this wonderful music using alternate tunings.
Personally I think regular repeating-scale tunings (where the scale is the same in every octave) are better for Hijaz than the likes of Richter, PowerBender etc, which have a different blow-draw pattern in each octave. They can certainly be used of course, but it's less confusing if you can use the same moves/breath patterns in each octave.
HIJAZ SCALES in POWER-CHROMATIC TUNING
As noted, PowerChromatic gives some good options for Hijaz, without requiring overblows. Using an L13 PC in G for explanation, the harmonica tuning is:
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BLOW G A C E G A C E G A C E G HOLE 1 2 3 4 5 6 7 8 9 10 11 12 13 DRAW A B D F# A B D F# A B D F# A
3rd POSITION (A root, starting holes 1/5/9):
draw -> drawbend -> blow/drawbend -> blow/drawbend -> blow/draw.
4TH POSITION (E root, starting blow 4/8):
blow/drawbend -> drawbend/draw -> draw -> blow/draw -> blow.
(You can substitute the blow A in the adjacent hole (holes 2/6/10 blow) in these positions, which can be handy).
6th POSITION (F# root, starting 4/8 draw):
draw -> blow -> drawbend/draw -> drawbend/draw -> blow/draw.
7th POSITION (C# root, starting holes 3/7 drawbend:
drawbend/draw -> drawbend/draw -> drawbend/draw -> draw -> drawbend.
12th POSITION (C Root, starting blow 3/7):
blow/drawbend -> blow/drawbend -> blow/drawbend -> drawbend -> blow.
Edvin showed earlier the variation of PowerChromatic tuning I've used on half-valved chroms for some Balkan tunes in Hijaz. In this case I retuned a PowerChromatic chrom in G, playing with a D root, where the E blow note is lowered to D#. It's good for fast play and slide decorations because only one bend is needed in the slide-out position, for the flat 6th note (Bb).
However, with the slider in, you get a cool alternate version of the scale using all draw notes and bends. I use both versions in my tune "Sweet Bulgarity", which is mostly in Hijaz:
The intro and the middle section are played with slide-in style. You can hear the switch-over from slide-out style in the 4th part of the tune, playing the same notes but with different feeling, at 2:04. After some rhythm playing the slide-in style resumes at 2:30 and lasts till 3:08.
The reason it works is because the slide-in notes are actually the same as they would be for a standard PowerChromatic half-valved chromatic in G with the slide pushed in. Thus in the video, when playing slide-in position, essentially I'm playing a G# diatonic harp in PowerChromatic tuning, with the root in D. (On a Lucky 13 G PowerChromatic it will come out in C#, as shown above).
This is exactly the expressive 7th Position sound that Edvin mentioned earlier...
All of the positions above have their good points, but probably 7th Position is the most 'Middle Eastern' sounding.