  Let L denote the variety of lattices. In 1982, the second author proved that
L is strongly tolerance factorable, that is, the members of L have quotients in
L modulo tolerances, although L has proper tolerances. We did not know any
other nontrivial example of a strongly tolerance factorable variety. Now we
prove that this property is preserved by forming independent joins (also called
products) of varieties. This enables us to present infinitely many {strongly}
tolerance factorable varieties with proper tolerances. Extending a recent
result of G.\ Cz\'edli and G.\ Gr\"atzer, we show that if V is a strongly
tolerance factorable variety, then the tolerances of V are exactly the
homomorphic images of congruences of algebras in V. Our observation that
(strong) tolerance factorability is not necessarily preserved when passing from
a variety to an equivalent one leads to an open problem.
