A gaussoid is a combinatorial structure that encodes independence in probability and statistics, just like matroids encode independence in linear algebra. The gaussoid axioms of Lněnička and Matúš are equivalent to compatibility with certain quadratic relations among principal and almost-principal minors of a symmetric matrix. We develop the geometric theory of gaussoids, based on the Lagrangian Grassmannian and its symmetries. We introduce oriented gaussoids and valuated gaussoids, thus connecting to real and tropical geometry. We classify small realizable and non-realizable gaussoids. Positive gaussoids are as nice as positroids: they are all realizable via graphical models. a 12|3 maps to σ 12 σ 33 − σ 13 σ 23 whereas a 13|2 maps to −(σ 12 σ 23 − σ 13 σ 22 ). Maintaining this sign convention is important to keep the algebra consistent with its statistical interpretation.Let J n denote the ideal generated by all homogeneous polynomials in the kernel of the map above. This defines an irreducible variety V (J n ) of dimension n+1 2 in the projective space P 2 n−2 (4+( n 2 ))−1 whose coordinates are P ∪ A. There is a natural projection from LGr(n, 2n) onto V (J n ), obtained by deleting all minors that are neither principal nor almost-principal. This is analogous to [30, Observation III.12], where the focus was on principal minors p I .Proposition 1.1. The degree of the projective variety of principal and almost-principal minors coincides with the degree of the Lagrangian Grassmannian. For n ≥ 2, it equals degree(V (J n )) = degree(LGr(n, 2n)) = n+1 2 ! 1 n · 3 n−1 · 5 n−2 · · · (2n − 1).

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