This "approximately unbiased test" is the "multiscale bootstrap" reported by Shimodaira in 2002, as another answer noted. The short answer is that simple bootstrapping does not always correctly determine the probability that a multi-dimensional observation came from a particular region of interest, as the shapes of boundaries between regions can affect simple bootstrap-estimated results. The multiscale bootstrap is one solution to this problem.
A motivating scientific issue was how to choose among competing phylogenetic trees of organisms, based for example on differences in DNA sequences. With only 4 possible bases (T, C, G, A) at each position along a DNA sequence, a simple model of conversion probabilities over time for each pair of bases allows for calculating the likelihood of a particular phylogenetic tree, given the DNA sequences of the various species in question.
The question is whether phylogenetic tree $i$ with log-likelihood $Y_i$ has the largest expected value of log-likelihood ($\mu_i$) among all $M$ trees under consideration. The vector $Y$ containing log-likelihood values for all $M$ trees is distributed around the parameter vector $\mu$ of corresponding expected log-likelihood values. That is, the hypothesis $H_i$
$$H_i : \mu_i \ge \mu_j, j=1,...,M,$$
represents a region in the $M$-dimensional parameter space $\mu$ having observed value $Y$.
This is the problem (Shimodaira 2002, p. 495):
Considering that $Y$ is distributed around $\mu$, one might believe that the hypothesis $\mu \in H_i$ is probable when the event $Y \in H_i$ is observed. There is, however, the possibility that $Y \in H_i$ by chance, even though $\mu \in H_j$ for some $j \neq i$. In other words, the selected tree with the largest $Y_i$ value is not necessarily the tree with the largest $\mu_i$ value.
This is a particular example of The Problem of Regions discussed by Efron and Tibshirani in 1998, following up in part on earlier work by Efron et al on bootstrapping phylogenetic trees. Even in a simple 2-region case, the probability of $Y \in H_i$ as estimated by bootstrapping from a sample that provided the maximum observed $Y_i$ (called the "confidence value," $\tilde\alpha$, by Efron and Tibshirani) is not the same as the confidence level $\hat\alpha$ with respect to a null hypothesis, $P_{1-\hat\alpha}(Y \notin H_i)$, unless the boundary between the regions is flat. Their Figure 2, copied below, shows the situation for a smooth curved boundary and sampling from a $K$-variate normal distribution with unknown means and identity covariance matrix:
The null hypothesis in this example is $\mu \in R_1$. Testing the probability that the observed $y=\hat\mu$ might have resulted from that null hypothesis requires knowing distances between $y=\hat\mu$ and the $\mu \in R_1$, information that is not sufficiently provided by simple bootstrapping from the sample unless the boundary between the regions is flat. In the example above with a boundary curved away from the observed value, many $\mu \in R_1$ are generally far from $y=\hat\mu$ so they are less likely to have contributed to finding the observed value by chance. A boundary curved toward $y=\hat\mu$, in contrast, would increase the chance of having found that value under the null hypothesis.
Efron and Tibshirani showed that a second bootstrap around the boundary point $\hat\mu_0$ closest to the observed point $y=\hat\mu$ provides a good approximate correction for curvature of the boundary, which combined with the "confidence value" $\tilde\alpha$ from the initial bootstrap gives a bootstrap-based estimate of the confidence level $\hat\alpha$.
Shimodaira found a different way to correct for boundary curvature, one that did not require finding the closest boundary point $\hat\mu_0$. He showed that taking bootstrapped re-samples with replacement of a size $N_s$ smaller or greater than the size $N$ of the data sample could serve the same purpose. As a different re-sample size necessarily increased or decreased the size of the corresponding bootstrap-estimated "confidence regions" around the observed $y$, it would also sample more or less from within the region $R$ of the null hypothesis, depending on the curvature of the boundary. This is illustrated in the following Figure from a later paper on the topic:
Here, $\tau^2 = N/N_s$. Shimodaira showed that coefficients of a simple nonlinear regression of bootstrapped $z$-values against $\tau$ and $1/\tau$ for a set of $\tau$ values both above and below 1 provide a correction for boundary curvature similar to that proposed by Efron and Tibshirani. This approach is also suggested to work reasonably well in situations with non-smooth boundaries.
The pvclust package in R implements this multiscale bootstrap test for hierarchical trees in general.
