What are the module categories over the modular tensor category Fib of Fibonacci anyons?
By Ostrik's work, we know these module categories correspond to separable algebras in Fib. I do not believe such things have been classified.
Davydov and Booker show there are no nontrivial commutative separable algebras in Fib, but I do not see that they make a clear statement for this more general case, without commutativity.
My guess is that there are indeed no nontrivial module categories for Fib.
There is only one equivalence class of indecomposable module categories, namely the trivial one.
Let us look into the possible algebras. They are $1$ and $1\oplus \tau$, and both have a unique algebra structure. For the first this is trivial and for the second is basically saying that the $A_4$ subfactor is unique. But $1\oplus\tau =\tau\otimes \bar\tau$ so it coincides with the internal action hom $\underline{\mathrm{Hom}}(\tau,\tau)$ of Fib as a module category over itself.
Here is another proof. The only possible connected étale (commutative) algebra is $1$, as the OP points out. Then by Corollary 3.8 in
Davydov, A.; Nikshych, D.; Ostrik, V.: On the structure of the Witt group of braided fusion categories. Selecta Math. (N.S.) 19 (2013), no. 1, 237–269.
module categories are in one-to-one correspondence with braided autoequivalences of $\mathrm{Fib}\to\mathrm{Fib}$ for which there exist only the trivial one.
