  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
TrFl$_{\boldsymbol{r},n}^{\geq 0}$ equals the nonnegative flag Dressian
FlDr$_{\boldsymbol{r},n}^{\geq 0}$, and that the points $\boldsymbol{\mu} =
(\mu_a,\ldots, \mu_b)$ of TrFl$_{\boldsymbol{r},n}^{\geq 0} =$
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.
