The conquest: complete or not
Joshua takes the whole land and leaves no survivors; Judges lists the territories Israel could not drive out — the two books' pictures of the same settlement.
Joshua 10:40-42
40So Joshua struck all the land, the hill country, the South, the lowland, the slopes, and all their kings. He left no one remaining, but he utterly destroyed all that breathed, as Yahweh, the God of Israel, commanded.
41Joshua struck them from Kadesh Barnea even to Gaza, and all the country of Goshen, even to Gibeon.
42Joshua took all these kings and their land at one time because Yahweh, the God of Israel, fought for Israel.
Judges 1:27-36
27Manasseh didn’t drive out the inhabitants of Beth Shean and its towns, nor Taanach and its towns, nor the inhabitants of Dor and its towns, nor the inhabitants of Ibleam and its towns, nor the inhabitants of Megiddo and its towns; but the Canaanites would dwell in that land.
28When Israel had grown strong, they put the Canaanites to forced labor, and didn’t utterly drive them out.
29Ephraim didn’t drive out the Canaanites who lived in Gezer, but the Canaanites lived in Gezer among them.
30Zebulun didn’t drive out the inhabitants of Kitron, nor the inhabitants of Nahalol; but the Canaanites lived among them, and became subject to forced labor.
31Asher didn’t drive out the inhabitants of Acco, nor the inhabitants of Sidon, nor of Ahlab, nor of Achzib, nor of Helbah, nor of Aphik, nor of Rehob;
32but the Asherites lived among the Canaanites, the inhabitants of the land, for they didn’t drive them out.
33Naphtali didn’t drive out the inhabitants of Beth Shemesh, nor the inhabitants of Beth Anath; but he lived among the Canaanites, the inhabitants of the land. Nevertheless the inhabitants of Beth Shemesh and of Beth Anath became subject to forced labor.
34The Amorites forced the children of Dan into the hill country, for they would not allow them to come down to the valley;
35but the Amorites would dwell in Mount Heres, in Aijalon, and in Shaalbim. Yet the hand of the house of Joseph prevailed, so that they became subject to forced labor.
36The border of the Amorites was from the ascent of Akrabbim, from the rock, and upward.
What each tradition says
Rabbinic / Midrashic
Hirsch notes that in Joshua xiii.–xix. many fragments largely parallel Judges i. These make it appear that the conquest was a slow, laborious process, with the tribes acting without concerted plan and nowhere under united central command. He assigns these fragments to J.
Emil G. Hirsch (1851-1923), The Jewish Encyclopedia, vol. 7, s.v. 'Joshua, Book of' (New York: Funk & Wagnalls, 1901-1906) — section on the composition of the book (source analysis)
Patristic
Augustine says the promise of God concerning the land of Canaan—that it extend from the river of Egypt to the great river Euphrates—was not fulfilled by Joshua after driving out the nations and dividing the land, nor in the whole time of the judges. It was not still prophesied as to come, but its fulfillment was expected.
Augustine of Hippo (354-430), The City of God (trans. Marcus Dods), in Nicene and Post-Nicene Fathers, ser. 1, vol. 2 (Edinburgh: T&T Clark, 1871) — Book XVII, ch. 2 ('At What Time the Promise of God Was Fulfilled Concerning the Land of Canaan')
Catholic
Drum states that the opening summary of Judges (i,1–ii,5) gives details whose historical value is attested by Joshua. He cites Juda's siege of Dabir, the Jebusites in Jerusalem, the Chanaanite in Gazer with Ephraim, and the Chanaanite dwelling with Manasses.
Walter Drum (1870-1927), The Catholic Encyclopedia, vol. 8, s.v. 'Judges' (New York: Robert Appleton Company, 1910) — section 'Historicity / External evidence'
Orthodox
No sourced position found. Checked the PD Greek/Eastern corpus: John of Damascus's Exposition (NPNF 2-09) and the 1672 Synod of Jerusalem (Confession of Dositheus) — no treatment of the Joshua/Judges question; Chrysostom's Hebrews homilies (NPNF 1-14) read Heb. 4:8 ('if Joshua had given them rest') only of the heavenly rest, not of the conquest's extent; Origen's allegorical Joshua homilies survive in English only in modern copyright translations. Honest gap.
Reformed / Protestant
Matthew Henry says Judah's lot was pretty well cleared of the Canaanites, but not thoroughly. The mountain Canaanites were driven out, but those in the valley with iron chariots held their ground; Judah failed, and the other tribes followed that cowardice rather than their other courage.
Matthew Henry (1662-1714), An Exposition of the Old and New Testament, vol. 2 (Joshua to Esther) (London, 1706-1710) — Exposition of Judges 1:19 ('Conquests of Judah')
Evangelical
Keil and Delitzsch explain that taking the whole land does not mean every town and village was conquered or every Canaanite driven out. Rather, Canaanite power was broken, their dominion overthrown, and the land thoroughly given to Israel, with those remaining reduced to powerless fugitives.
Carl Friedrich Keil (1807-1888) and Franz Delitzsch (1813-1890), Biblical Commentary on the Old Testament: Joshua, Judges, Ruth (trans. James Martin) (Edinburgh: T. & T. Clark, 1875) — Commentary on Joshua 11:16-23 ('Subjugation of the Whole Land')