It is called a T-Sylvester equation, or *-Sylvester equation in the complex case. Solvability conditions and a pseudocode algorithm based on the Schur form are in https://doi.org/10.13001/1081-3810.1479 . Analogous considerations for a more general class of equations and a Fortran-90 implementation of the resulting solution algorithm are in my paper https://doi.org/10.1002/nla.2261 . I don't think you will find something in Slicot, because it has no immediate control theory applications.
Federico Poloni
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