If we include a genetic relatedness matrix in our analyses (e.g., using linear mixed regression), is it still necessary to control for population structure using principal components in addition? Or does the GRM itself already provide sufficient control for population structure?

In theory the GRM should be enough to control for population structure, as it can be seen as fitting N PCs.

But I donâ€™t think is is a problem to fit PCs in a Linear Mixed Model, and in some GWAS software (e.g. BOLT-LMM) fitting some of the top PCs can improve computational speed (Mixed model association for biobank-scale data sets - PMC).