We introduce a phase space with spinorial momenta, corresponding to fermionic derivatives, for a $2d$ supersymmetric $(1,1)$ sigma model. We show that there is a generalisation of the covariant De Donder-Weyl Hamiltonian formulation on this phase space with canonical equations equivalent to the Lagrangian formulation, find the corresponding multisymplectic form and Hamiltonian multivectors. The covariance of the formulation makes it possible to see how additional non-manifest supersymmetries arise in analogy to those of the Lagrangian formulation.

We then observe that an intermediate phase space Lagrangian defined on the sum of the tangent and cotanget spaces is a first order Lagrangian for the sigma model and derive additional supersymmetries for this.}