Base-phi converter time!
Retina, 81 80 bytes

K`0.
"$+"{+`(1(\.)|(1\.(0?1)*0?)1)(00|$)
${3}0${2}11
0\.
1.
+0`0?1(\.?)1
10${1}0


Try it online! Works by repeatedly adding \$ 1 \$. Explanation:

K`0.


Start off with \$ 0. \$.

"$+"{`


Repeat the number of times given by the input.

+`(1(\.)|(1\.(0?1)*0?)1)(00|$)
${3}0${2}11


If the \$ \phi^0 \$ bit is set, use the identity \$ \phi^0 = \phi^{-1} + \phi^{-2} \$ to move bits away until there is enough room.

0\.
1.


Add \$ 1 \$.

+0`0?1(\.?)1
10${1}0


Minimise the number of bits by reversing the above identity.