I posted about this in 2018:
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Albert Guerard's summary of the two constitutions of 1946:
"The constitution elaborated by the Assembly under the inspiration of two men of keen intelligence, Andre Philip and Pierre Cot, [Phillip was a Socialist, Cot was elected as a republican but was known by everyone to be close to the Communists--in fact, the Venona documents make it clear that he was a KGB agent--DT] was a bold departure from the Orleanist compromise of 1875. It reverted to the tradition of the old Radicals, so different from the trimmers and time-servers who had later adopted the name. The Senate was abolished outright. The President became an even more shadowy figure than under the Third Republic. The single and omnipotent Assembly elected and could remove the prime minister. It was a Jacobin constitution and paved the way for the autocracy of a majority party. It was passed by a coalition of the Communists and Socialists over the opposition of the M.R.P.'s. General de Gaulle, now a private citizen, pronounced against it. When on May 5, 1946, it was submitted to a referendum, it was rejected by 10,583,724 votes to 9,453,675.
"A new Constitutional Assembly was elected on June 2, 1946. There was no sensational change; but since the M.R.P. was now the largest party, Bidault became president-premier with Gouin, Socialist, and Thorez, Communist, as vice-premiers. The second constitution diverged widely from the first but not, as De Gaulle had expected, in the direction of the American system, with a strong and independent executive. It was purely and simply a rehash of the Constitution of 1875, with a figurehead president, a senate renamed Council of the Republic, elected in a very complicated fashion, and a popular chamber or national assembly. Both constitutions were prefaced with elaborate declarations of rights. Both made provisions for a French Union or Commonwealth to supersede the colonial empire.
"Again De Gaulle expressed his disapproval. On October 13, 1946, 9,120,576 Frenchmen endorsed the new regime, and 7,980,333 rejected it. But there were 25,379,917 registered voters: the nine million "yeas" represented only 36 per cent of the electorate. The constitution was evidently a compromise, unloved even by its sponsors. The M.R.P. in particular hastened to say that it was voted to end a provisional situation fraught with discomfort and dangers, but that it was in need of prompt and drastic amendment. So, with perverted logic, France progressed from the provisional to the precarious. It might have been wiser to do without a permanent constitution for a few years longer, or even, like the France of the Ancient Regime and like England to the present day, to dispense with a written constitution altogether..."
https://archive.org/stream/franceamodernhis006433mbp#page/n475/mode/2up
The fear of some conservatives that the first constitution would lead to an eastern-European-style "people's democracy" in France seems overblown. True, as long as the Communist and Socialists had a majority in the National Assembly they could theoretically govern by themselves under this constitution. However, the Socialists (SFIO) rejected all attempts by the PCF to make them into a junior partner. They insisted that the government include the Christian Democratic MRP as well as Communists and Socialists. If this was true even in 1945-6 when the left-wing postwar tide was at its height, it would certainly be true after the Cold War intensified in 1947. Guy Mollet, the SFIO leader, famously said that the PCF was "not on the Left but in the East."
https://en.wikipedia.org/wiki/French_Section_of_the_Workers'_International I think the PCF will still be ejected from the government as it was in OTL. And in any event, the PCF-SFIO majority in the National Assembly would not last forever.
However, it wouldn't solve the Fourth Republic's problems, either. Having a Council of the Republic (as the Senate was renamed) did nothing to solve France's problems, but it didn't cause them, either. Nor did the Fourth Republic Presidents (Auriol and Coty). When the generals revolt, the Assembly will still call on De Gaulle to come to power and he will still write a new constitution..