IDA fails to solve when encountering algebraically inconsistent `u` after callback

[topolarity]

Extracted from SciML/OrdinaryDiffEq.jl#2127

IDA also appears to struggle to get past the first stop:

julia> using DiffEqCallbacks, Sundials, Plots
julia> f(du, u, p, t) = [du[1] - u[2], u[2] - p * sin(t)]
julia> prob = DAEProblem(DAEFunction(f), [0.0, 0.0], [0.0, 0.0], (0., 2π), 1,
                         callback=PresetTimeCallback(0:.3:2π, integ->(integ.p=-integ.p;)),
                         differential_vars=[true,false])
julia> sol = solve(prob, IDA())
julia> plot(sol; xlim=(0, 2π))

Originally posted by @topolarity in SciML/OrdinaryDiffEq.jl#2127 (comment)

[topolarity]

Output:

[IDAS ERROR]  IDASolve
  At t = 0.3 and h = 2.38735e-08, the error test failed repeatedly or with |h| = hmin.

retcode: Unstable

@ChrisRackauckas chimed in already to help:

IDA has a reinitialization. It must not be reinitializable with your choice of initialize. Brown basic?

I don't know why this problem wouldn't be re-initializable, though. It's extremely smooth (linear in u[2]).