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I'm late by a year, but just in case, the result from this one-page paper seems to answer your question (but it has nothing to do with the Wada property):

M. Brown and J. M. Kister, Invariance of complementary domains of a fixed point set, Proc. Amer. Math. Soc. 91 (1984), no. 3, 503–504. MR 744656

The theorem is as follows: Let f $f$ be a homeomorphism of a connected topological manifold M $M$ with fixed point set F. $F$. Then either (1) each connected component of M-F $M-F$ is invariant, or (2) there are exactly two components and f $f$ interchanges them.
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I'm late by a year, but just in case, the result from this one-page paper seems to answer your question (but it has nothing to do with the Wada property):

M. Brown and J. M. Kister, Invariance of complementary domains of a fixed point set, Proc. Amer. Math. Soc. 91 (1984), no. 3, 503–504. MR 744656

The theorem is as follows: Let f be a homeomorphism of a connected topological manifold M with fixed point set F. Then either (1) each connected component of M-F is invariant, or (2) there are exactly two components and f interchanges them.