Here's an argument for an irregular octahedral-prism die:

> Have two faces be regular octagons with edge-length $1$
>
> Have the prism height be $h$
>
> Call the $8$ rectangular prismatic faces with labels "1" through "8"
>
> Call the octahedral faces "9" and "10" or "top" and "bottom"

It may require physical simulation to find the critical heights I am defining below, and it will obviously be a function of the material used to make the die , the surface onto which it is rolled, and of the frictional coefficients $\mu_{die}$ and $\mu_{felt}$, and the nature of the rolling mechanism used to toss the die onto the felt (most likely the tumbling action in 6-dimensional state-space describing the initial velocity and rotational movement of the die and the height at which it is released, etc.)


Because of symmetry, we can say


$p_{top}=p_{bottom}=x,$ 

$p_{i\in(1,8)}=y$,

$ 2x+8y=1$,

$0 \le x \le 1 \textrm{ and } 0 \le y \le 1$

There will be a critical height $h_{min}$, where for $h \lt h_{min}$, 
the bias for "top" and "bottom" will be greater than for the prismatic faces, and $x \gt y$.



There will also be a critical height $h_{max}$ where because the long axis is substantially longer than the flat octahedral faces, the momentum will be such as to cause a die landing on an octahedral face "top" or "bottom" to continue to move and fall/topple onto a rectangular prismatic face, with each of the 8 prismatic faces equally likely.

Somewhere between the critical maximal and minimal heights will be a fair height, $h_{fair}$, where this non-regular die will roll fairly onto any of the ten faces with $$p=\frac{1}{10}=0.1$$

Now this is a rotationally symmetric die with 8-rotations around the long axis and 2 orientations of the long axis, so it's not the "assymetric die" you've asked for, but it could be the beginning for a similar construction of an asymmetric die.