This is going to be a loose thread of examples of Marx criticizing phrenology and other examples of "race science." This is done not with the intention to completely vindicate him for his racist tendencies, but simply to complicate the picture some have of him as a total bigot.

i repeat, do not use this thread to say Marx was innocent of all racial and national biases. He was not. This is simply a corrective to what i feel is the dominant narrative that he was some sort of 'race realist' white supremacist.

An early example of Marx criticizing the social nature of race can be found in 1847's "Wage Labour and Capital." While he does not explicitly criticize race as a concept in general, he does criticize the idea that race is a determining factor of social position, class, etc.

From the chapter "The Nature and Growth of Capital," "What is a Negro slave? A man of the black race. The one explanation is worthy of the other. A Negro is a Negro. Only under certain conditions does he become a slave."
https://www.marxists.org/archive/marx/works/1847/wage-labour/ch05.htm

1852's "The Heroes of the Exile!" contains a passage in which Marx mocks phrenology proponent Gustav Struve by doing a fake "analysis" of his features (this work is largely a satirical one), and he concludes by comparing phrenology to necromancy.
https://www.marxists.org/archive/marx/works/1852/heroes-exile/ch04.htm

A letter to Engels in 1856 includes an oddly insightful section in which Marx talks about how colonization helps to bolster the "middle class" and discusses also how the normal route by which serfdom comes about is by an imposed "racial dualism."
https://www.marxists.org/archive/marx/works/1856/letters/56_10_30.htm

(As an aside: I do not think he presents a truly cohesive viewpoint in this letter, as it is more of a brainstorm than anything. However, i do think Marx's discussion of colonialism as "conquest and racial dualism" is interesting when compared with Fanon's "Manichaeanism".)

1868, a letter is sent to Kugelmann which says "One of my friends here, who dabbles a lot in phrenology, said yesterday when looking at the photograph of your wife: A great deal of wit! So you see, phrenology is not the baseless art which Hegel imagined."
https://www.marxists.org/archive/marx/works/1868/letters/68_01_11b.htm

Much like the bit on Struve, Marx is using phrenology ironically, mockingly, and concludes by insulting it. This is also important, as it shows Marx read Hegel's critique of phrenology (it is in the Phenomenology, so it was already reasonable to assume he did).

The next couple examples will come from Marx's "Ethnological Notebooks," written from 1880 to 1882, just a year before he died (1883).
https://www.marxists.org/archive/marx/works/1881/ethnographical-notebooks/notebooks.pdf

From section 162, Marx criticizes Henry Maine for saying that research had validated the theory of racial difference, and says the science really shows how people develop in different societies. "Dieser asinus bildet sich ein dass" roughly means "This ass/idiot imagines that."

Section 164, he continues to critique Maine, now along different lines. Firstly, he criticizes him for making it seem as if the social organization of the Irish at the time was a consequence of their race. Marx calls him a "blockheaded Englishman" and says this is "Foolishness."

Second, Maine creates a racial binary, where the "Aryan race" develops landed property by splitting individuals off from the collective while other races develop landed property by raising individuals (Chiefs) above the collective. Marx calls this out as a false dichotomy.

First, because these two ultimately mean the same thing. Landed property emerges as private property emerges from collective property. Second, because the mechanisms Maine describes for each can be found in the other, meaning some Aryans develop like non-Aryans, & vice versa.

When Maine speaks of the Aryans, Marx interjects "who?" and proceeds to throw doubt on the category entirely. After all, the French, British, Romans, etc, all developed in quite different ways. What role does "race" really play in development then?

Throughout all of Marx's comments on Maine, we see him repeatedly bring up the latter's inability to deal with social relations instead of individual people(s). Marx mocks him over and over for leaning on supposedly innate racial categories instead of social ones.

In section 187, while on one of those rants, Maine begins again to talk about "The Aryans" and Marx inserts "the devil take this 'Aryan' cant!" And criticizes Maine for using race to explain social relations, both in terms of property and government.

There are others throughout his notes, and the introduction to the collection even explicitly states "A recurrent theme is Marx’s systematic and uncompromising rejection of race, racism and biologism generally as a determinant without further qualification of social affairs."

i might post other examples later, but for now, regardless of whatever biases Marx may have had, he was nonetheless right when he said that "the emancipation of the productive class is that of all human beings without distinction of sex or race."
https://www.marxists.org/archive/marx/works/1880/05/parti-ouvrier.htm