Here is a Linear Relaxation of the perfect matching polytope, which is integral for simple graphs:
(1) For each $v \in V, \sum_{e \in \delta(v)} x_e = 1.$
(2) For each $S \subseteq V, |S|$ is odd, $\sum_{e \in \delta(S)} x_e \geq 1.$
Does the integrality of this polyhedron also hold for graphs with self-loops? (where $\delta(S)$ is the set of edges with exactly one endpoint in $S$, note that this includes self-loops on vertices in $S$).