A straightforward argument reveals that a taut foliation is Reebless, and of course there are many examples of Reebless foliations that are not taut. I guess that there are many examples of three-manifolds having a Reebless foliation but not a taut one.

Looking for an example I found the following one. The 3-manifold obtained by doing $+37/2$-surgery along the $(-2, 3 ,7)$-pretzel knot contains a Reebless foliation (essentially because it contains an incompressible torus whose complement consists of two copies of the complement of the trefoil knot) but does not contain a taut foliation since it is a Heegaard Floer L-space ($+18$-surgery along $K$ gives a lens space, and an L-space is forever!*).

So my main questions are:

Is there a classical proof (i.e. that does not make use of Heegaard Floer homology) of the fact that the $+37/2$ surgery along $K$ does not support a taut foliation?

Can someone give me other (in some sense simpler) examples of three-manifolds having a Reebless foliation but not a taut one?

Are there examples in which the use of Heegaard Floer homology can't be apparently bypassed?

*i.e. if $S^3_n(K)$ is an L-space, then so is $S^3_q(K)$ for every rational number $q\ge 2g(K)-1$.

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