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We investigate positivity and probabilistic properties arising from the Young-Fibonacci lattice $$\mathbb{YF}$$, a 1-differential poset on binary words composed of 1's and 2's (known as Fibonacci words). Building on Okada's theory of clone Schur functions (Trans. Amer. Math. Soc. 346 (1994), 549-568), we introduce clone coherent measures on $$\mathbb{YF}$$ which give rise to random Fibonacci words of increasing length. Unlike coherent systems associated to classical Schur functions on the Young lattice of integer partitions, 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 $$\mathbb{YF}$$. We connect Fibonacci positivity with total positivity of tridiagonal matrices, Stieltjes moment sequences, and orthogonal polynomials in one variable from the ($$q$$-)Askey scheme. Our second family of results concerns the asymptotic behavior of random Fibonacci words derived from various Fibonacci positive specializations. We analyze several limiting regimes for specific examples, revealing stick-breaking-like processes (connected to GEM distributions), dependent stick-breaking processes of a new type, or discrete limits tied to the Martin boundary of the Young-Fibonacci lattice. Our stick-breaking-like scaling limits significantly extend the result of Gnedin-Kerov (Math. Proc. Camb. Philos. Soc. 129 (2000), no. 3, 433-446) on asymptotics of the Plancherel measure on $$\mathbb{YF}$$. We also establish Cauchy-like identities for clone Schur functions (with the right-hand side given by a quadridiagonal determinant), and construct and analyze models of random permutations and involutions based on Fibonacci positive specializations and a version of the Robinson-Schensted correspondence for $$\mathbb{YF}$$.