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In this web page, a photo from a repertory of photos from the Spanish Civil War, at the website of the archive of Italian Istituto Luce, shows a rally with Francisco Franco within a theatre. From the balconies several flags hang, among which are the German and the Italian ones, and then one with what seems a light-coloured Star of David on a darker background.

Image source: Istituto Luce / Archivo Luce

Does anyone know whose nation or organisation was this flag, and why it was shown at a Francoist meeting?

Most probably because this is not really a star of David, but a hexagram. As such it has probably less to do with Jews, Zionists or Israel.

Whether this is an occult symbol, popular with fascists, or a stylised version of an etoile, six pointed star or estrella de seis puntas, the basic pattern is widespread and used in heraldry for cities, where this meeting could have taken place.

Example for six-points (plus: no connection to Islam, despite the crescent) Armañanzas

Example for a Spanish nationalist or falange fighter, no connection to Jewish symbolism:

(src)

That said, in this case it might simply be a reminder of the glorious origins of the falangist forces of Franco. And that is in North-Africa. The Spanish Protectorate of Morocco sported this type of flag in various variations:

Source: WP List of Moroccan flags

But compare the apparently argent variant with Spanish Protectorate of Morocco (1912-1956) and

The symbol of armed authority was a hexagram … Starting from the reign of the Sultans Hisham (1790/’92-’97) and Suleyman (1792-1822) the hexagram is uninterruptedly documented on coins until 1956.

But if that sounds still like a tough sell to Catholic Spanish falangistas, either something looking like a Jewish or a Moroccan symbol, then it might be as well recognising a symbol of Mary or stella maris:

Just like the Star of David, the Star of Mary is a Six-Pointed Hexagram

A 13th century Church in Asturias, Spain known as 'Iglesia de Santa María' (1270 A.D.)

Among the medas and insignia used in connection to the Army of Africa, the Order of the Medhauia seems to stick out:

To reward those very distinguished and notorious facts of Spaniards and Moroccans carried out in the Zone of Morocco entrusted to the Protectorate of Spain.
(a) (b)
<sub>(a) Rifaat (equivalent to gentleman)
Visirial decree of 8 of safar of 1345 -18 August 1926-
Award period: 1926-1956
Compose the central part of the badge of an enameled circle, in which the setting Sun is represented, that is to say "The Mogreb", reflected in a blue-green sea. Circumscribed to this circle is a strip, also of enamel and of Prussian blue color, limited by a golden fillet. Everything described above is enclosed in a Solomonic ring.
The ribbon on which the decoration hangs is green, with a white central stripe.

On the back, in a circle of white enamel, read the following Arabic inscription: "Muley el Mehdi Ben Ismail Ben Mohamed, glorify him God"
Both the Solomonic ring and the ornaments will be silver. It is suspended from the left side of the chest

(b) Sumú-u (equivalent to the Great Cross)
Visirial decree of 8 of safar of 1345–18 August 1926-
Award period: 1926-1956
Compose the central part of the badge of an enameled circle, in which the setting Sun is represented, that is to say "The Mogreb", reflected in a blue-green sea. Circumscribed to this circle is a strip, also of enamel and of Prussian blue color, limited by a golden fillet. Everything described above is enclosed in a Solomonic ring.
On the back, in a circle of white enamel, read the following Arabic inscription: "Muley el Mehdi Ben Ismail Ben Mohamed, glorify him God"
Identical to the previous one and attached to the left side, taking, in addition, a band that crosses the chest from right to left and from whose end the second class badge hangs</sub>

And further campaign militaria offered by wwwmilitariabcncom:

Arm badge Moroccan Army Corps. 92 Division. Nationalist Army. Spanish Civil War.

Collar badge. Supervisor of the Mehala Jalifiana . (Colonial Troops). Nationalist Army.. Spanish Civil War.

And finally, a propaganda poster from that time in colour:

Venus Dictator in Furs. Painting by Paco Ibera

_{Franco in furs like Bette Davis. He wears his Miss España sash in the traditional Spanish colors (as opposed to the red, yellow & purple of the Republican flag) and on his chest is a maltese cross (an order of knighthood perhaps? – this forum leads me to the Grand Cross of the Order of San Fernando). Below the cross is a pin of the symbol of the Falangists. His hand rests on a cane while his other hand holds his hat; a garrison cap with a gold tassel. He tries to strike a regal pose. Franco alone takes up nearly half of the image.}

_{On the lower left is a cartouche with Franco's well known and brief declaration of the end of the Spanish Civil War. In a Roman font it declares in large type "La Guerra ha terminado" (The war is over). There is probably no connection between this and John Lennon's song "War is Over."}

One of the flags in the background is either a pentagram or a hexagram flag of the caliph of Morocco.

The hexagram flag is almost certainly a reference to Spanish Morocco (with which, as noted by LangLangC in his answer, Franco and his forces had a strong connection) and the Army of Africa. The Army of Africa played a key role for Franco's Nationalists during the Spanish Civil War.

These sets of stamps were issued, probably in 1939, with Franco and the hexagram.

FRANCO, VICTORIOSO (VICTORIOUS)

Image source: ebay

FRANCO, CAUDILLO (LEADER)

Image source: ebay

FRANCO, GENERALISSIMO

Image source: ebay

The ebay seller dates them 1937-39, but this flickr comment specifying 1939 seems more likely:

This was issued by Spanish Morocco in 1939 to aid disabled soldiers in
North Africa. On the back the price was una peseta.

The hexagon also appears on the emblem of the 2nd Regulares Company of Tabor Rif II

Tabor (Regiment) Rif II was founded in 1911 by Lieutenant Colonel
Berenguer to fight in Africa.

The use of the six-pointed star is a peculiarity of these units. This
star appeared in the old Moroccan silver coins, stamps and documents,
and it was adopted to reflect the origin of the (formerly) Moroccan
native troops recruited for these units, mostly coming from the Rif
Mountains.

Image source: crwflags.com

This link provided by SJuan76 shows other emblems with hexagons.

SJuan76 has also commented that "A Moorish Guard (Guardia Mora) was kept for his [Franco's] personal security until the independence of Morocco" and has provided this additional link.