I was inspired by France's constituencies for residents overseas
Using 2000 data, there are about 1.17 million natives of Washington living in the rest of the country. In 2000, the ideal Washington legislative district was about 120,000 people. This would result in about ten abroad districts at approximately 117,000 each. Two states had enough Washingtonians to warrant more than one district. They were Oregon and California, which had about 1.6 and 2.1 seats worth each, respectively. I used 1995-2000 migration data to get an approximate idea of where Washingtonians generally moved, and redistricted accordingly. These are the only two states I did this for.
District 1 (Red) - Portland & Northwest Oregon: 117,370
Unsurprisingly the smallest in land area of the ten districts, given its proximity to WA and the shared Portland-Vancouver metro.
District 2 (Light green) - Rural Oregon, Alaska, and Hawaii: 118,946
Proximity to Oregon again. Only about 11k Washingtonians in Hawaii, but WA has a strong connection with Alaska (about 5% of Alaskans are born in WA).
District 3 (Blue) - Northern California: 114,762
Quite a bit of movement between the Pacific states, no surprise. Presumably a large majority of this is in the Bay Area, but I didn't check.
District 4 (orange) - Southern California: 117,234
Second smallest in terms of land area.
District 5 (purple) - Inland Northwest: 117,307
Another shared metro (Spokane-Couer d'Alene), as well as strong connections in general with ID and MT. While this technically contains SD and NE, the vast majority of the population (about 110k) is in ID+MT+WY, which is why I named it Inland NW.
District 6 (pink) - Southwest: 116,151
Close to half is Arizona, quite a few retirees down there (no surprise). Also, CA's extra tenth of a district (it was 2.1, remember) was shifted here by moving San Bernadino County.
District 7 (brown) - South Central: 118,474
About half in Texas. not too surprising given what a large state it is. Another 30k+ in CO.
District 8 (yellow) - Midwest: 115,636
We've gotten far enough away that the districts are getting pretty big in land area, especially considering their overall population. The Washingtonian population isn't especially concentrated in any one state.
District 9 (dark green) - South: 115,912
Same as above. Worth noting abnormally large number of Washingtonians in VA, due to DC confusion.
District 10 (dark red) - Northeast: 113,914
Same as above. Again, strangely large number of Washingtonians reported in DC and MD, presumably because of idiots who don't know the difference between the state and the city. Even with this artificial boost, this is a pretty big region considering how populous it is.
The numbers don't quite match up. Part of this is certainly due to Puerto Rico (and possibly other territories) that I excluded. I also may have made an error.