Probably you will not get a successful third party called the Socialist party. Could you get a Labor or Farmer-Labor Party that would be as "socialist" as the British Labour Party or Norwegian Labor Party? That would first of all require support from organized labor. That is not inconceivable; there was a strong opposition Socialist bloc in the AFL led by Max Hayes.
https://en.wikipedia.org/wiki/Max_S._Hayes If Hayes coud triumph over Gompers, and both major parties nominate conservative candidates, such a party could have some prospects.
I realize that this is before your POD but I have always been fascinated by the prospect of the Populist Party evolving into America's leading socialist party, with Eugene Debs as leader (again this requires the Democrats nominate a conservative instead of Bryan). IMO Populists had more in common with European socialist parties than many people think. Consider the following letter of Colorado's Populist Governor Davis H. Waite to the American Federation of Labor national convention in Denver in 1894. Note that he quotes Karl Marx's son-in-law Paul Lafargue!
https://en.wikipedia.org/wiki/Paul_Lafargue
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The following communication from Governor Waite was received and read:
State Capitol, Denver, Dec. i4, i894.
Gentlemen of the Convention:
Official business has compelled me to decline the invitation to make you a personal address, but at the request of your committee I send you a brief letter in relation to the remedy which, in my judgment should be adopted by all labor organizations to accomplish their purposes. I say nothing in relation to organization which all admit to be indispensable, or the ordinary mode of operations which, especially in Great Britain, have been so successful in keeping up wages, preventing child labor and ameliorating the conditions of the workingmen. The labor unions of the United States undoubtedly can learn much practical good from the experience of their brethren in Europe. I was pleased to see in the remarks of Brother Burns at the Wednesday meeting the exhortations to members of trade unions to "get together." It seems that in England, although in that country political means are more difficult than here, workmen or representatives of labor are elected to office, and exert great influence in the British Parliament. In the United States, political means and remedies are much more easily accessible, and legislation under greater control than in the old country, and yet the legislation of the United States in favor of labor cannot be compared with the legislation in the more advanced of other nations. The great object of government is the protection of the citizen in life, liberty and the pursuit of happiness, yet the legislative councils of every state and of Congress are monopolized for the protection of invested capital, the guaranteeing of profits and the creation of fictitious corporations which may dominate the individual. Instead of guarding the rights of the individual citizen, legislation has been monopolized to grant special powers to combinations. For the past thirty years the wealth of the nation has been a more important factor in the body politic than the rights of the people. As the result of the trend of such legislation, the prosperity of the nation has been destroyed, and wealth is concentrated in the hands of the few. To benefit this few who are consumers, prices are forced below the cost of production, wages are kept to the lowest possible rate, and whole nations are fast being reduced to pauperism.
Said Lafargue, a member of the French Assembly, in a recent speech:
"When capitalism visits a country it is as if a hurricane had broken loose that tears down and destroys everything that stands in its way--men, animals, the quick and the dead.
"When European capital let itself down in Egypt it seized the fellahs with their beasts of burden, their wagons and their prongs, as so many blades of grass, and carried them off to the Isthmus of Suez; with its iron hand bent them under the yoke of servitude, and there, scorched by the sun, worried with hunger and thirst, attacked by fever, the bones of thirty thousand of those victims whitened the canal. Capital seizes upon free and healthy strong and happy people, and immures them by the hundreds of thousands in the mills, the factories and the mines; when it lets them go again they are prematurely old, scrofulous, anaemic, consumptive."
God knows, the condition of the laboring man needs amelioration, pity, charity; but the laboring men need justice more. If we are to remove evil, common sense demands that we should strike at the cause; it is useless to busy ourselves with alleviating fever and plague, and leave the foul source of disease undisturbed.
The capitalism which controls our legislation, which dominates our national conventions, and dictates political platforms and policies, within the past few years has almost defiantly overridden the constitutional provisions, national and state, protecting civil rights. Four years ago Benjamin Harrison, Republican President of the United States, used the military power of the United States in Wyoming to protect a band of land monopolists, who murdered two settlers upon the public lands, and the blood of those murdered men today cries from the ground in vain. The same capitalism, acting through Grover Cleveland, Democratic President, used the military power of the United States in compelling labor to perform work on railroad lines, sending the military power into Illinois, without any request of the legislature thereof or of the Governor of the State, and contrary to the Constitution of the United States. Within the last four years United States Courts, overriding the Constitution of the United States, have arrested without warrant, tried without a jury and convicted without evidence, sending workmen to prison for contempt, of a so-called receiver of a railroad who is really not so much an officer of the United States as the post-master at the country cross-roads.
A recommendation is now before Congress from the Commander-in-Chief of the United States Army, General Schofield, recommending an increase of the army, its withdrawal from frontier posts and its concentration near the large cities for the purpose of overawing laboring men, putting down strikes and compelling laboring men to work on the railroad lines of capitalism.
How else can organized labor resist the inroads upon personal rights, which attacks come by political means, not only through unfriendly legislation, but through the plain usurpations of executive power?
I am aware that the present policy of the Trades Union is non-partisan, and I recognize fully your right to adopt such policy as you may approve. I would treat your action with due respect, but conceding the fact that capitalism controls our legislative bodies, that the United States Senate is simply a hospital for millionaires, that our rights are swept away, not only by Congress and Legislatures, but by the usurpation of the judiciary, there is but one remedy, and that is the workingmen must "get together' politically, and support that party which in good faith declares for the rights of humanity as against the rights of property.
Yours respectfully,
DAVIS H. WAITE
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https://books.google.com/books?id=t_rTAAAAMAAJ&pg=RA1-PA35