  A Dirac structure is a Lagrangian subbundle of a Courant algebroid,
$L\subset\mathbb{E}$, which is involutive with respect to the Courant bracket.
In particular, $L$ inherits the structure of a Lie algebroid. In this paper, we
introduce the more general notion of a pseudo-Dirac structure: an arbitrary
subbundle, $W\subset\mathbb{E}$, together with a pseudo-connection on its
sections, satisfying a natural integrability condition. As a consequence of the
definition, $W$ will be a Lie algebroid. Allowing non-isotropic subbundles of
$\mathbb{E}$ incorporates non-skew tensors and connections into Dirac geometry.
Novel examples of pseudo-Dirac structures arise in the context of quasi-Poisson
geometry, Lie theory, generalized K\"ahler geometry, and Dirac Lie groups,
among others. Despite their greater generality, we show that pseudo-Dirac
structures share many of the key features of Dirac structures. In particular,
they behave well under composition with Courant relations.
