  We study the challenging thermal phase transition to stripe order in the
frustrated square-lattice Ising model with couplings J1<0 (nearest-neighbor,
ferromagnetic) and J2>0 (second-neighbor, antiferromagnetic) for g=J2/|J1|>1/2.
Using Monte Carlo simulations and known analytical results, we demonstrate
Ashkin-Teller criticality for g>= g*, i.e., the critical exponents vary
continuously between those of the 4-state Potts model at g=g* and the Ising
model for g -> infinity. Thus, stripe transitions offer a route to realizing a
related class of conformal field theories with conformal charge c=1 and varying
exponents. The transition is first-order for g<g*= 0.67(1), much lower than
previously believed, and exhibits pseudo first-order behavior for g* < g < 1.
