I think the possibility that the "Republicans" might have somehow been unified and moderate enough early on to win over sufficient positive support from Britain and make the French feel more confident in positive support has been explored and rejected adequately enough. The Spanish Republic, such as it was, was obviously not a comfortable liberal nation rudely conquered by alien fascist thugs; it was a nation with a very thin grasp on liberal democracy because a whole lot of Spanish were much more radically inclined than any government in London would have been comfortable supporting. And as someone said above, these people expressing their radically divergent visions of how life in Spain should go is part of what we call "democracy."
So for positive support for the actually existing democracy, which was hardly coherently united, we'd need a radical power interested in seeing radical alternatives to the liberal order of the 1930s evolve such as was happening in Spain.
People want a Bastard? What else was Stalin trying to be?
To be fair--I suppose Stalin probably didn't care much what happened to Spain and his actual interest in aiding the Republicans was to put Soviet hardware and military doctrine to the test, picking up a subservient Spain if he could get it but if not, writing off the Spanish.
So if the question is, "Could the Republicans have won?" with a POD of 1936, I'm afraid it looks unlikely to me.
If we can relax the POD and suppose that a more reasonable Soviet Union, or some kind of Red Russia (such as the one in LordInsane's Central East timeline) or some radical power somewhere existed that could counter the Axis support and organize a unified, or at least coordinated, battle plan with the resources they contribute, that might have done the trick. Upthread at post 10 NothingNow
proposed something like that if there had been a union of Latin American left-wing regimes--Mexico, Cuba, Chile say. Obviously they could only have offered a shoestring of help but their presence might have catalyzed more cooperation among the "Republican" factions? But to oppose the hardware and troops the Italian Fascists and Nazis threw into the ring there probably needed to be more substantial help. And the best one could hope from the British government was that it could be persuaded to be truly neutral and not covertly actually aiding Franco as I am grateful
someone has already pointed out they did.
A radical Russia would not have been trusted by Britain's Tories nor considered safe allies for the insecure French progressives, but I can see them winning enough respect among the democratic citizens of these parliamentary governments to at least win a stay on active opposition, provided the Russians conducted themselves less cynically than Stalin ordered OTL. A leftist intervention that wanted to win would have had to do better at building alliances and been less concerned with winning hegemony at the cost of ultimate victory. LI's Red Russia of the Central East timeline is not a Bolshevik dictatorship but a somewhat queasy coalition of everyone to the left of the Tsarists, in a much-truncated rump Russia that however did avoid much of the devastation of the Great War and the Civil War OTL--they lost a lot of land but kept the
people, and the infrastructure of the core they hung onto, and as a coalition would have more plausibly fielded an expedition that would be committed to supporting all anti-fascist factions provided they were willing to work together for mutual defense.
Now obviously the more broadly based a Red Russia is, and the more successful, the more it is likely to panic reactionary elements in all countries, so it is not clear how ITTL the Russians would be able to get help to Spain. Of course ITTL it is also not clear how reactionary the governments of either Italy or Germany would be nor whether there would be any sort of fascist power to back Franco anywhere. Perhaps that timeline would be a good one to explore a Spanish Civil War where no foreign governments get involved, or at any rate modest, furtive backing of Franco or his analog can be countered by volunteers alone?
A POD where at any rate Stalin is not calling the shots in the Kremlin, or his position is much weaker and he has real diversity in Soviet policymaking to contend with and fields a force with the mission of winning the war rather than a long-shot attempt to rule Spain, might be much later in time than LordInsanes and involve a victorious Entente instead.