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We define a notion of nonassociative $\mathrm{L}^p$-space associated to a$\mathrm{JBW}^*$-algebra (Jordan von Neumann algebra) equipped with a normalfaithful state $\varphi$. In the particular case of $\mathrm{JW}^*$-algebrasunderlying von Neumann algebras, we connect these spaces to a complexinterpolation theorem of Ricard and Xu on noncommutative $\mathrm{L}^p$-spaces.We also make the link with the nonassociative $\mathrm{L}^p$-spaces of Iochumassociated to $\mathrm{JBW}$-algebras and the investigation of contractivelycomplemented subspaces of noncommutative $\mathrm{L}^p$-spaces. More precisely,we show that our nonassociative $\mathrm{L}^p$-spaces contain isometrically the$\mathrm{L}^p$-spaces of Iochum and that all tracial nonassociative$\mathrm{L}^p$-spaces from $\mathrm{JW}^*$-factors arise as positivelycontractively complemented subspaces of noncommutative $\mathrm{L}^p$-spaces.

Nonassociative $\mathrm{L}^p$-spaces and embeddings in noncommutative $\mathrm{L}^p$-spaces