Don't know about a "Red Israel" but a Communist-friendly one is a possibility -- until they intellectually moved more towards a "thinking" socialism rather than the Soviet "thuggist" model.
Maybe Britain withdraws from the Middle-East pre 1945 so Egypt, Lebanon & Jordan (1950's tourism continuing) would remain pro-British (no Suez war) while Israel becomes pro-Soviet -- just to keep everyone on opposite sides

Many of Israel's intellectual elite in social, military, government & artistic areas came from the Kibbutz movement -- started in 1910 by Communist Jews fleeing Tsarist religious persecution (pre-Communism). In 1920 the political arm of this movement gains power ....
A Socialist govt. would probably have prevented the Israel / Palistine split & may have avoided the "settlers" moving on Gaza today -- Israel / Palistine would still be one country, not two -- depriving neighbouring Arab states of the current main reason to become involved. ((Moslem / Christian / Jewish religious conflicts only erupted within Israel post-1947 due to the creation of the State of Israel, previously all concerned lived peacefully together in the same streets.))
So fewer Mid-East wars ? More diplomacy ? Israel wouldn't have such a large defence industry, the technical ability that has spawned the Merkava Tank & missile-guidance systems (used in all US air-air missiles) would have gone into other areas like medical technology.
On the military side -- being pro-Communist, the IDF would be supplied Soviet weaponry ? -- JS-3Ms then T-55 & T-62 - but with a bigger / better turret.
The IDF actually found that the T-62's 125mm gun was more accurate & over greater distances (on the range) than the Centurions/M-60s 105mm -- but only with a well trained & practised crew, so, in an ambush situation, the IDF could sit their Tiran 6's on a small rise & pick off the enemy at will. The IDF did experimentally apply Blazer armour to their Tirans & they'd still design their Achzarit (T-55-based) APC.
Egypt, Lebanon & Jordan would have Centurions (later upgraded to Oliphant 1 & 2 standard) then Challengers with the Teledyne / Jordanian turretless 130mm gun & auto-loader (turret-crew now in hull) -- Jordan currently calls this still-experimental vehicle the "Al Hussein" MBT with "Falcon 2" turret.
Presume Isreal would initially get Yak-9s & Sturmoviks, Mig-15/17, later Mig-29s or Su-33s etc for the Air Force and modern Soviet patrol boats & subs too ?
Egypt, Lebanon & Jordan get Meteors, Buccaneer, Jaguar etc ?
There was a thread on this over on What-If Modellers in the Ideas Bank but I can't find it now.