By Schaefer's dichotomy theorem, this is NP-complete.
Consider the case where all clauses have 2 or 3 literals in them; then we can consider this as a constraint satisfaction problem over a set $\Gamma$ of relations of arity 3. In particular, the relations $R(x,y,z)$ are the following: $x \lor y$, $x \lor \neg y$, $\neg x \lor \neg y$, $x \oplus y \oplus z$, $x \oplus y \oplus \neg z$.
Now apply Schaefer's dichotomy theorem, in its modern form. Check each of the six operations to see if they are a polymorphism:
- Unary 0: Not a polymorphism of $x \lor y$.
- Unary 1: Not a polymorphism of $\neg x \lor \neg y$.
- Binary AND: Not a polymorphism of $x \lor y$. (Consider $(0,1,0)$ and $(1,0,0)$; they both satisfy the relation, but their pointwise-AND $(0,0,0)$ doesn't.)
- Binary OR: Not a polymorphism of $\neg x \lor \neg y$. (Consider $(0,1,0)$ and $(1,0,0)$; they satisfy the relation, but $(1,1,0)$ doesn't.)
- Ternary majority: Not a polymorphism of $x \oplus y \oplus z$. (Consider $(0,0,1)$ and $(0,1,0)$ and $(1,0,0)$; they satisfy the relation, but their majority $(0,0,0)$ doesn't.)
- Ternary minority: Not a polymorphism of $x \lor y$. (Consider $(0,1,0)$, $(1,0,0)$, and $(1,1,0)$; they satisfy the relation, but their minority $(0,0,0)$ doesn't.)
It follows that this problem is NP-complete, even if you restrict all the XOR clauses to be of length at most 3.
On the other hand, if all the XOR clauses are restricted to be of length at most 2, then this is in P. In particular $(x \oplus y)$ is equivalent to $(x \lor y) \land (\neg x \lor \neg y)$, so any such formula is equivalent to a 2SAT formula, whose satisfiability can be determined in polynomial time.