Two small additions.
(1) Ordinal number of Stiemke's doctorate since the start of the Berlin mathematics faculty in the 1810s. According to p.361 of [K.-R. Biermann, Die Mathematik und ihre Dozenten an der Berliner Universität 1810-1933. Akademie-Verlag Berlin 1988. ISBN 3-05-500402-7], a book which is out of print, Stiemke was the 161st to obtain a doctorate in mathematics from University of Berlin (there was only one such before WW2). (The first was one Samuel Ferdinand Lubbe in 1818.) Notably, Biermann lists July 1, 1925 as the date of Stiemke's doctorate, while he lists 16 July 1914 for the date of the oral exam, with an exclamation mark (presumably to draw attention to the delay between exam and doctorate). Again, the ordinal number 161 depends on Biermann's decision to take the publication date of Stiemke's dissertation as the date of the doctorate. Whether this was more than an arbitrary decision by Biermann is not apparent from the book (I checked the index of loc. cit.; at the few places where Stiemke appears in the text, nothing more is said than what is already in this thread.) In particular, nothing is said about who was the driving force behind a poshumous doctorate eleven years after the `fact'; presumably it was Emmy Noether, but that is my guess only.
(2) Stiemke's bibliography. Stiemke has exactly three published articles:
(2.1913) Stiemke, E. Sur les modules dénombrables. (French) C. R. 157, 273-274 (1913).
(2.1915) Stiemke, E. Über positive Lösungen homogener linearer Gleichungen. Math. Ann. 76, 340-342 (1915). ([eudml](https://eudml.org/doc/158695}, DOI: 10.1007/BF01458147)
(2.1926) Stiemke, E. Über unendliche algebraische Zahlkörper. Mathematische Zeitschrif 25, 9-39 (1926). (eudml, DOI: 10.1007/BF01283824)
By the way, (2.1915) contains an interesting little result giving a sufficient condition for the existence of a solution to a homogeneous system of linear equations such that each component of the solution is positive; the result was new to me (I bet it is a special case of something more modern, but I didn't manage to make such a connection):
Theorem (Stiemke 1915). For any $(n,m)\in\mathbb{N}^2$ and $A\in\mathbb{R}^{n\times m}$, if for every $v\in\mathbb{R}^n$ the vector $v^{\mathrm{t}}A$ is either the zero vector or has at least one negative component, then there exists $x\in\mathbb{R}^m$ having all components strictly positive such that $Ax=0$.
