Base-phi converter time! The Golden Ratio Bureau is obsessed with this new thing they call base-phi. You see it and decide to code-golf, as is your natural instinct.

To be specific, base-phi is a number system like base 10, except it uses the number phi, or the golden ratio, as its base. A ones digit would be x*10^y, but in base phi 10 is replaced with phi. Base phi also uses 2 digits, 0 and 1.

Your goal is to accept input that is a base 10 positive natural number, then treat it as such and convert it to base phi.

Due to base phi being able to represent all numbers in more than one way, your program should convert input to its "minimal" representation. This is the representation with the least 1 digits.
Output can have trailing characters but only if the language must output them with no circumvention.

Do not use any built-ins for base conversion. You may use a built-in for phi, but the base conversion should rely on string manipulation and other mathematical operations.

Your program must support inputs up to 2147483647, or lower, depending on your language's limit for integers.

Your program can deal with any undefined behavior as you wish.

Testing cases for accuracy can be done at this link. In case you do not wish to use the link, here are the numbers 1-15 in base phi.

1 = 1
2 = 10.01
3 = 100.01
4 = 101.01
5 = 1000.1001
6 = 1010.0001
7 = 10000.0001
8 = 10001.0001
9 = 10010.0101
10 = 10100.0101
11 = 10101.0101
12 = 100000.101001
13 = 100010.001001
14 = 100100.001001
15 = 100101.001001


The shortest program following these rules wins. Have fun.
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.