ACROSS
 1. F ALTERS outside ↺OUT SHELL↺ = FALLOUT SHELTERS <bomb defences>
 9. BE MUSES = BEMUSES <puzzles>
13. <Algonquian spirit> MANITOU = (.a)MAN I TOU(t.)
14. A M_ B ROSE BURNS + IDE = AMBROSE BURNSIDE <general with mutton-chops>
15. <monuments> TRIUMPHAL ARCHES = I inside TRUMP HA + LARCHES
16. ←MOD RAT'S← = STARDOM <celebrity>
17. <Polish area> SILESIA = S_ I_ L_ E_ S_ I_ A_
18. <chemical property> MOLECULAR WEIGHT = MOLE CU + AR WE inside LIGHT
19. HANS EAT GAELIC* + ←EU← = HANSEATIC LEAGUE <trade union>
23. STING outside IF = SIFTING <using riddle?>
25. <playwright> ROSTAND = RO[-man] + STAND
26. (LET ETONIAN)* outside OAR IS = NEO-ARISTOTELIAN <philosophical>
27. <one-armed bandits here> AMUSEMENT ARCADE = A M USE MEN + TAR CADE
28. <African emperor> SEVERUS = SEVER US
29. <evangelists> GIDEONS = O inside GI DENS
30. (TUDOR ASS IN TUNIC)* = TITUS ANDRONICUS <Shakespeare play>
31. RHINE SS outside [-l]IVERWORT = RIVERWORTHINESS <fluvial suitability>
36. (RAGE outside H) after NU = NURAGHE <Sardinian tower>
39. I NO outside LIMB = IN LIMBO <not yet decided>
40. OFFS H_ (HIS FOREIGN)* = OFFSHORE FISHING <trawler work?>
41. R inside (DOUBLE + TAN DADS) = DOUBLE STANDARDS <unfair rules>
42. NEE + RIDE* = NEEDIER <having less money?>
43. (YES GAME)* = EYEGASM <great viewing experience>
44. S REALISES* STANCE = SALES RESISTANCE <unwillingness to buy>

DOWN
 1. <these carry disease> FOMITES = ←OF← MITES
 2. <coastal region> LONG ISLAND SOUND = LONG IS L AND SOUND
 3. OP TIMES = OPTIMES <Cambridge mathematicians>
 4. TRUSS outside EH (RATE EDEN)* = TRUE-HEARTEDNESS <loyalty>
 5. <Holland city> HAARLEM = HAM outside _A_ R_L_E_
 6. (BOILER ACT)* outside ←MEDLAR← = LIBERAL DEMOCRAT <politician>
 7. <fashionably green> ECO-CHIC = (CHOC-ICE)*
 8. S HELSI_ _L VERSE outside T + IN = SHEL SILVESTEIN <author>
 9. <tough> BRUISER = B←C_, CRUISER
10. MAN (DATE OF HAVEN) outside _E = MANDATE OF HEAVEN <government authority>
11. SH INDIG[-nation] = SHINDIG <party>
12. <&lit> SPERMATOGENESIS = (PERSON'S GAMETES)* outside _I_
19. <violin from Norway> HARDANGER FIDDLE = HARD ANGER FIDDLE
20. E_ YALE TO F RUM ELI _A = EYALET OF RUMELIA <Balkan province>
21. <walks> CONSTITUTIONALS = CONSTITUTION AS outside L_
22. G-MAN'S outside ER + HEP + HERD'S = GERMAN SHEPHERDS <beastly guards>
23. <root problems> SHOESTRING FUNGI = SHOESTRING FUN _GI_
24. ←SIRI← RE INCOGNITO* = IRIS RECOGNITION <biometrics>
32. <scavenger> VULTURE = V [-c]ULTURE
33. O (SOUP'S)* M_ = OPOSSUM <marsupial>
34. <one lacking faith> INFIDEL = (IN FIELD)*
35. <internal membranes> SEROSAE = ←SORES← + A+E
37. <admire> RESPECT = REST outside PEC
28. <make swollen> ENGORGE = ENG ORG _E