Well, the high Atlantic component is not your only difference to the average AJ. You are also more West Asian, and less North Sea / Baltic / Eastern Euro (see my addition to your table above). That would suggest that you have taken up very little ancestry from Eastern Europe, and some Sephardim ancestry instead.
But even a AJ/ Sephardim mix wouldn't make you that "Atlantic", Sephardim also have just 14% of it. Actually, there aren't many populations that are high on "Atlantic" but at the same time low on "North Sea". The best here is French Basques (45% Atlantic vs. 17% North Sea) - in terms of a relevant, ancient Jewish community that should mean Narbonne.
The "best fit" population, as you probably already have realised, is West Sicilians. But West Sicilians, and even more so French Basques lack the West Asian component and are too Western Mediterranean. So I looked for a population that is high on "West Asian" but at the same time low on "Eastern European" and "Siberian". Best fit here is Georgian Jews. From the three together one can already pretty well emulate your ancestry structure, but they miss an East African & Red Sea component. Yemenite Jews fill the gap perfectly. That's not meaning you have actually Yemenite Jewish ancestry, they should rather compensate for the specifically Jewish element that is missing within French Basques. So here is my result:
Population | Georgian Jewish | West Sicilian | Yemenite Jewish | French Basque | Mix | John Doe |
| 35.6% | 26.1% | 9.1% | 29.2% | 100% | |
North_Sea | 2.27 | 9.76 | 0.27 | 16.85 | 8.30 | 8.44 |
Atlantic | 2.67 | 18.32 | 1.05 | 45.40 | 19.08 | 19.52 |
Baltic | 1.63 | 4.70 | 0.16 | 4.79 | 3.22 | 4.00 |
Eastern_Euro | 2.91 | 3.27 | 0.11 | 2.82 | 2.72 | 1.99 |
West_Med | 6.95 | 17.76 | 4.40 | 25.04 | 14.82 | 14.60 |
West_Asian | 35.22 | 10.64 | 5.41 | 0.83 | 16.05 | 16.32 |
East_Med | 36.66 | 26.75 | 54.11 | 2.74 | 25.76 | 26.20 |
Red_Sea | 8.06 | 5.37 | 27.46 | 0.73 | 6.99 | 6.79 |
South_Asian | 2.77 | 0.57 | 0.48 | 0.13 | 1.22 | 0.00 |
Southeast_Asian | 0.17 | 0.19 | 0.06 | 0.28 | 0.20 | 0.62 |
Siberian | 0.10 | 0.08 | 0.10 | 0.04 | 0.08 | 0.00 |
Amerindian | 0.16 | 0.01 | 0.08 | 0.03 | 0.08 | 0.00 |
Oceanian | 0.11 | 0.52 | 0.18 | 0.10 | 0.22 | 0.29 |
Northeast_African | 0.29 | 1.33 | 6.11 | 0.17 | 1.06 | 1.24 |
Sub-Saharan | 0.04 | 0.72 | 0.01 | 0.04 | 0.21 | 0.00 |
Except for the South Asian component that comes with the Georgian Jews, it fits quite nicely.
Otherwise, you appear to be a living proof of Jewish settlement in Eastern Europe (Galizia) during Khazar times.
My family does have a tradition that AJs descend from Sephardis who came to Poland after the expulsion, my mum is indeed darker than my biological dad (my mum looks Mediterranean, she was mistaken for a local in Italy, my biological dad was fair, dark blonde, green/blue eyes, I got the fairness from him, I'm dark blonde, green eyed and fair skinned, I was mistaken for an Anglo-German).
As for the French Basque, well look at my K15 results:
Admix Results (sorted):
# Population Percent
1 East_Med 26.2
2 Atlantic 19.52
3 West_Asian 16.32
4 West_Med 14.6
5 North_Sea 8.44
6 Red_Sea 6.79
7 Baltic 4
8 Eastern_Euro 1.99
9 Northeast_African 1.24
10 Southeast_Asian 0.62
11 Oceanian 0.29
Single Population Sharing:
# Population (source) Distance
1 Ashkenazi 9.1
2 Sephardic_Jewish 9.48
3 Tuscan 10.37
4 Greek 10.57
5 North_Italian 16.43
6 Bulgarian 16.92
7 Cyprian 17.15
8 Turkish 18.23
9 Romanian 18.75
10 Lebanese_Muslim 19.17
11 Syrian 20.2
12 Serbian 21.68
13 Spanish_Andalucia 22.16
14 Spanish_Extremadura 22.31
15 Spanish_Murcia 22.54
16 Jordanian 23.15
17 Portuguese 23.16
18 Samaritan 23.47
19 Spanish_Valencia 23.63
20 Spanish_Castilla_Y_Leon 23.85
Mixed Mode Population Sharing:
# Primary Population (source) Secondary Population (source) Distance
1 64.8% Lebanese_Muslim + 35.2% French_Basque @ 2.51
2 57.8% Lebanese_Muslim + 42.2% Spanish_Aragon @ 3.25
3 55.9% Lebanese_Muslim + 44.1% Spanish_Castilla_La_Mancha @ 3.28
4 51.9% Assyrian + 48.1% Spanish_Aragon @ 3.35
5 55.3% Lebanese_Muslim + 44.7% Spanish_Valencia @ 3.43
6 53.7% Lebanese_Muslim + 46.3% Spanish_Andalucia @ 3.48
7 59.4% Lebanese_Muslim + 40.6% Southwest_French @ 3.52
8 52.3% Spanish_Andalucia + 47.7% Assyrian @ 3.54
9 50% Spanish_Castilla_La_Mancha + 50% Assyrian @ 3.58
10 50.6% Spanish_Valencia + 49.4% Assyrian @ 3.72
11 59.3% Assyrian + 40.7% French_Basque @ 3.81
12 58.6% Lebanese_Muslim + 41.4% Spanish_Cantabria @ 4
13 51.9% Spanish_Murcia + 48.1% Assyrian @ 4.38
14 54.2% Lebanese_Muslim + 45.8% Spanish_Murcia @ 4.4
15 55.7% Lebanese_Muslim + 44.3% Spanish_Castilla_Y_Leon @ 4.43
16 53.5% Assyrian + 46.5% Southwest_French @ 4.44
17 56% Lebanese_Muslim + 44% Spanish_Cataluna @ 4.47
18 63.8% Syrian + 36.2% French_Basque @ 4.5
19 52.7% Assyrian + 47.3% Spanish_Cantabria @ 4.69
20 50.4% Spanish_Castilla_Y_Leon + 49.6% Assyrian @ 4.71
As for the Khazar admixture, maybe, but that still wouldn't explain so much Atlantic admixture. Also I highly doubt I have Yemenite or even Georgian Jewish ancestry, in the case of the former, I'd expect much more red sea and a much closer proximity to Palestinians, Jordanians, Bedouins and Saudis, with Yemenite Jews being Judaised Arabs and all.