We study positivity and probabilistic properties arising from the Young--Fibonacci lattice $\mathbb{YF}$, a 1-differential poset on binary (Fibonacci) words of 1's and 2's, graded by digit sum. Building on Okada's theory of clone Schur functions (Trans. Amer. Math. Soc. 346 (1994), 549--568), we define clone coherent measures on $\mathbb{YF}$ that generate random Fibonacci words of increasing length; unlike for the Young lattice (powered by the classical Schur functions), clone coherent measures are generally not extremal on $\mathbb{YF}$. Our first main result is a complete characterization of Fibonacci positive specializations -- parameter sequences which yield positive clone Schur functions on $\mathbb{YF}$. Second, we connect Fibonacci positivity with: (i) total positivity of tridiagonal matrices; (ii) Stieltjes moment sequences; (iii) the combinatorics of set partitions; and (iv) families of univariate orthogonal polynomials from the (q-)Askey scheme. We further link moment sequences of orthogonal polynomials to combinatorial structures on Fibonacci words, a connection that may be of independent interest. Third, we analyze scaling limits of the induced random words, obtaining stick-breaking-type limits (linked to GEM laws), new dependent stick-breaking limits, and limits supported on the discrete part of the Martin boundary of $\mathbb{YF}$. These results significantly extend the asymptotics of the Plancherel measure on $\mathbb{YF}$ proved by Gnedin--Kerov (Math. Proc. Camb. Philos. Soc. 129 (2000), 433--446). Finally, we prove Cauchy-type identities for clone Schur functions with quadridiagonal-determinant right-hand side (in contrast to the product form for classical Schur functions), and construct models of random permutations and involutions from Fibonacci-positive specializations together with a Robinson--Schensted correspondence adapted to $\mathbb{YF}$.