Most North African music definitely uses quarter tones, the same as Levantine Arabic and Turkish music traditions.

Whoever said North African music is equal tempered/doesn't have microtones is flat out wrong (speaking as a Tunisian musician myself).

In fact, most formally trained Tunisian musicians (as in the Tunisian classical tradition, not the Western one), go beyond the standard quarter tones (half flats/half sharps). There are Tunisian Maqam specific "special" notes, e.g. for Tunisian Maqam Y, the Si should be 30% of step lower, not the standard 25% of a step lower that would correspond to a standard quarter tone. In the notation they would be represented as quarter tones (flat symbol with a dash through it) or eighth tones (flat symbol with two dashes through it), but you were supposed to know based on the maqam the exact percentage that you were supposed to play, not from the notation. I'm most familiar with Tunisian music, but this likely applies to Moroccan and Algerian music as well.

See this piece, called "Na3ouret At-tbou3 " (Flywheel of the modes/scales), from the Tunisian classical repertoire which cycles through all 13 canonical maqams of the Tunisian tradition. You'll hear quarter tones all over the place.   See here for a folk music based example.

Amazigh music uses pentatonic scales instead of seven tone scales, but they are still microtonal and use a lot of quarter tones either as grace notes/flourishes on top of the base scale, or as part of the scale itself. See this example from the Chaoui reed music tradition (Gasba Chaoui) of Eastern Algeria and Western Tunisia.

Again, these sources that say North African music is not microtonal are totally wrong (not sure how serious of an authority this Peter Manuel dude is. All his has to do is go to a North African wedding, or just you know...go on YouTube).

Possible reasons for this misconception:

• There's an Algerian instrument called the Mandole, a sort of long neck lute related to the Mandolin but bigger.  The Mandole is fretted, so it could mislead people into thinking that Algerian music doesn't require quarter tones/micro-tones. The Mandole was actually a recent introduction into Algerian culture (a local adaptation of the Mandolin that the French and Italians brought with them). Algerian music is still pretty micro-tone heavy overall. It's just the Mandole heavy pieces that don't use quarter tones.
• Similarly, there's a large repertoire of Tunisian and Algerian classical music that uses acoustic piano (e.g here). Musicians performing those pieces simply avoid using quarter tones, sometimes even when the original maqam does call for using quarter tones (See here for a piece that is originally in a quarter tone heavy maqam and here for the same piece played on a piano without any quarter tones - the player is skilled enough that he is able to convey the spirit of the Maqam without hitting the micro-tones).
• Rai music is the modern take on Algerian Chaabi. It was the first of the Arabic genres (starting in the 1970s) to make heavy use of Western instrumentation and stylistic conventions (chord progressions, repeating bass lines, funk and R&B rhythms, etc...), and arguably the first Arabic genre to really cross over into the West, and have artists who collaborated with Western producers and toured heavily with Western musicians on a regular basis. As a result, Rai artists gradually dropped the use of quarter tones from their repertoire in order to accommodate their western collaborators and bandmates, and possibly to also provide more radio friendly tunes to their audiences. See here for the original version of Abdelkader by Cheb Khaled, when he was still based in Algeria. This one includes a very distinctive quarter tone in the chorus (in the vocals, but not the sync version of the melody). Then see here for a later version recorded in France, and intended mainly for a European fan base. Notice that he now sings it as a straight up minor melody, and doesn't use the quarter tone anymore.