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We present an action for $N=1$ supergravity in $10+2$ dimensions, containingthe gauge fields of the $OSp(1|64)$ superalgebra, i.e. one-forms $B^{(n)}$ with$n$=1,2,5,6,9,10 antisymmetric D=12 Lorentz indices and a Majorana gravitino$\psi$. The vielbein and spin connection correspond to $B^{(1)}$ and $B^{(2)}$respectively. The action is not gauge invariant under the full $OSp(1|64)$superalgebra, but only under a subalgebra ${\tilde F}$ (containing the $F$algebra $OSp(1|32)$), whose gauge fields are $B^{(2)}$, $B^{(6)}$, $B^{(10)}$and the Weyl projected Majorana gravitino ${1 \over 2} (1+\Gamma_{13}) \psi$.Supersymmetry transformations are therefore generated by a Majorana-Weylsupercharge and, being part of a gauge superalgebra, close off-shell. Theaction is simply $\int STr ({\bf R}^6 {\bf \Gamma})$ where ${\bf R}$ is the$OSp(1|64)$ curvature supermatrix two-form, and ${\bf \Gamma}$ is a constantsupermatrix involving $\Gamma_{13}$ and breaking $OSp(1|64)$ to its ${\tildeF}$ subalgebra. The action includes the usual Einstein-Hilbert term.

A locally supersymmetric $SO(10,2)$ invariant action for $D=12$ supergravity