Let E < U₀ .  
Define k² = (2m/ħ²)E, k₀² = (2m/ħ²)U₀, ρ² = (2m/ħ²)(U₀ - E) = k₀² - k².
For plane wave solutions we found a phase shift upon reflection.
Is this phase shift related to some observable effect?
Let us construct a wave packet using only  k < k₀, incident from the left.  Let g(k) = 0 for  k ≥ 0.  For each plane wave we have
A₁'/A₁ = (k - iρ)/(k + iρ) = [(k² + ρ²)^1/2exp(-iθ(k))]/[(k² + ρ²)^1/2exp(iθ(k))] = exp(-i2θ(k)),
with  tanθ(k) = ρ/k = (k₀² - k²)^1/2/k.

In region 1, at t = 0, we may write the wave packet as
Ψ(x,0) = (2π)^-1/2∫g(k) [exp(ikx) + exp(-i2θ(k)) exp(-ikx)]dk.
In region 1 U = 0, the wave packet represents a free particle. 
For a free particle we have  ω(k) = ħk²/(2m)  and  
Ψ(x,t) = (2π)^-1/2∫g(k) exp(i(kx - ω(k)t))dk + (2π)^-1/2∫g(k) exp(-i(kx + ω(k)t + 2θ(k))dk
or
Ψ(x,t) = (2π)^-1/2∫g(k) exp(i(kx - ω(k)t))dk + (2π)^-1/2∫g'(k') exp(i(k'x - ω(k)t - 2θ(k'))dk'
with k^’ = -k and g’(k^’) = g(k).  We again assume that g(k) has a pronounced peak at some k = k_i,. 
Then g’(k^’) has a pronounced peak at k ^’= -k_i. 
Let us choose g(k) to be real, a(k) = 0.  For the centers of the incident and the reflected wave packets we then have
x_i(0) = -dα(k)/dk|_ki = 0,  x_i(t) = dω/dk|_k=ki =  ħk_it/m,
x_r(t) = dω/dk'|_k'=-ki + 2dθ/dk'|_k'=-ki =  -ħk_it/m - 2dθ/dk|_k=-ki.
We have to evaluate dθ/dk.  We know tanθ in terms of k. 
With dtanθ/dθ = (1 + tan²θ) = k₀²/k² and dtanθ/dk = -(k₀² - k²)^1/2/k² we have
dθ/dk = -(k₀² - k²)^-1/2
Therefore  x_r(t) = -ħk_it/m + 2/(k₀² - k_i²)^1/2.

The incident and the reflected wave packet exist only in the region x < 0.  For times t << 0, x_i moves with constant velocity to the right, x_r is positive and does not lie in region 1, there is no reflected wave.  (The incident and reflected waves, by definition, only exist in region 1.)  For times t >> 0, x_i is positive and does not lie in region 1, there is no incident wave, x_r moves with constant velocity to the left.  But the reflection has introduced a time delay T.  We have
x_r(t) = -v(t - T),  v = ħk_i/m,  T = (2m/ħk_i)(k₀² - k_i²)^-1/2.
This time delay is a direct consequence of the phase shift that we found for each plane wave solution.  The particle is not instantaneously reflected.  This is in principle a measurable effect.

When a wave packet encounters a potential step U₀ and it contains plane wave components with E > U₀, then it is partially reflected and partially transmitted.  If it contains only plane wave components with E < U₀, then it is totally reflected.  But for some time ΔT the wave packet penetrates into the classically forbidden region.  This results in a time delay upon reflection.