Summary: A flag positroid of ranks \(\boldsymbol{r}:=(r_1<\dots<r_k)\) on \([n]\) is a flag matroid that can be realized by a real \(r_k\times n\) matrix \(A\) such that the \(r_i\times r_i\) minors of \(A\) involving rows \(1,2,\dots,r_i\) are nonnegative for all \(1\leq i\leq k\). In this paper we explore the polyhedral and tropical geometry of flag positroids, particularly when \(\boldsymbol{r}:=(a, a+1,\dots,b)\) is a sequence of consecutive numbers. In this case we show that the nonnegative tropical flag variety \(\operatorname{TrFl}_{\boldsymbol{r},n}^{\geq 0}\) equals the nonnegative flag Dressian \(\operatorname{FlDr}_{\boldsymbol{r},n}^{\geq 0}\), and that the points \(\mu=(\mu_a,\dots,\mu_b)\) of \(\operatorname{TrFl}_{\boldsymbol{r},n}^{\geq 0}=\operatorname{FlDr}_{\boldsymbol{r},n}^{\geq 0}\) give rise to coherent subdivisions of the flag positroid polytope \(P(\underline{\boldsymbol{\mu}})\) into flag positroid polytopes. Our results have applications to Bruhat interval polytopes: for example, we show that a complete flag matroid polytope is a Bruhat interval polytope if and only if its \((\leq 2)\)-dimensional faces are Bruhat interval polytopes. Our results also have applications to realizability questions. We define a positively oriented flag matroid to be a sequence of positively oriented matroids \((\chi_1,\dots,\chi_k)\) which is also an oriented flag matroid. We then prove that every positively oriented flag matroid of ranks \(\boldsymbol{r}=(a,a+1,\dots,b)\) is realizable.