  The skein algebra of a marked surface, possibly with punctures, admits the
basis of (tagged) bracelet elements constructed by Fock-Goncharov and
Musiker-Schiffler-Williams. As a cluster algebra, it also admits the theta
basis of Gross-Hacking-Keel-Kontsevich, quantized by Davison-Mandel. We show
that these two bases coincide (with a caveat for notched arcs in once-punctured
tori). In unpunctured cases, one may consider the quantum skein algebra. We
show that the quantized bases also coincide. Even for cases with punctures, we
define quantum bracelets for the cluster algebras with coefficients, and we
prove that these are again theta functions. On the corresponding cluster
Poisson varieties (parameterizing framed $PGL_2$-local systems), we prove in
general that the canonical coordinates of Fock-Goncharov, quantized by
Bonahon-Wong and Allegretti-Kim, coincide with the associated (quantum) theta
functions. Long-standing conjectures on strong positivity and atomicity follow
as corollaries. Of potentially independent interest, we examine the behavior of
cluster scattering diagrams under folding.
